Československá akademie věd Matematický ústav

TOEPLITZ OPERATORS ON BERGMAN-TYPE SPACES Miroslav Engliš

Kandidátská dissertační práce

Školitel: prof. RNDr. Vlastimil Pták, DrSc.

Praha, January 1991

CONTENTS

A SURVEY OF THE PRESENT STATE OF THE SUBJECT THE OBJECTIVE AND METHODS OF THIS THESIS RESULTS OF THE THESIS ADDENDUM Chapter 1. INTRODUCTION Chapter 2. DENSITY OF TOEPLITZ OPERATORS Chapter 3. TOEPLITZ ALGEBRAS Chapter 4. THE TOEPLITZ CALCULUS Chapter 5. THE BEREZIN TRANSFORM CONCLUSIONS FOR FURTHER DEVELOPMENT IN THE AREA LIST OF NOTATION REFERENCES

A SURVEY OF THE PRESENT STATE OF THE SUBJECT

The subject of this thesis are Toeplitz operators on Bergman-type spaces^{1}, an area which has received some attention during the last decade. These operators arise mainly in two areas of mathematics: in connection with quantum mechanics (the quantization procedures on symmetric spaces, cf. [5], [6]) and in the theory of subnormal operators (the basic model for a subnormal operator is the Bergman shift; see [9], [23]). Whereas the theory of these operators on the Hardy space H^{2} is by now well understood and has become classical (cf. e.g. [22] and the bibliography given therein), its area measure counterparts are far less tractable, and the results are sparser and often markedly different (see, for instance, [2] or [6]). It is natural to ask if this is not because the class of Toeplitz operators on a Bergman-type space is, in some sense, larger than on H^{2}. The classical Toeplitz operators are characterized by the intertwining relation S^{∗}T S = T (see f.i. [24]), and so form a w^{∗}-closed subset of B(H^{2}) of infinite codimension. In my paper [16], I have shown that such a characterization is impossible on A^{2}(D). In [17], it turned out that Toeplitz operators on A^{2}(D) are, in fact, dense in B(A^{2}(D)) in the strong operator topology, which gives an affirmative answer to the conjecture above. Further progress has been attempted by Gautrin [13], who proved even norm-density; unfortunately, his proof contained an error, and so the problem of norm-density of Toeplitz operators remained open. So did also the similar problem whether the C^{∗}-algebra generated by them contains all bounded operators (and if not, what is it?). The corresponding C^{∗}-algebra on the Hardy space is closely related to the theory of Toeplitz (or symbol) calculus (cf. [22], Appendix 4), the study of which has contributed to the development of powerful techniques of operator theory (dilation theory, commutant lifting theorem). In case the corresponding C^{∗}-algebra on the Bergman space contained all bounded operators, an analogue of such a calculus could be of exceptional interest. Some attempts in this direction have been made e.g. by McDonald and Sundberg (for symbols in a certain C^{∗}-subalgebra of L^{∞}(D); see [2] and the references therein). A closely related question of compactness of Hankel operators on Bergman-type spaces has been intensively studied by various authors — Axler et al. [3], Zhu, Stroethoff, Zheng, Berger and Coburn (see [6], [27]); these issues, however, are not discussed in this thesis.

THE OBJECTIVE AND METHODS OF THIS THESIS

The objective of this thesis is a solution to some of the problems mentioned above. It is proved that the norm closure of the set of all Toeplitz operators on a Bergman-type space contains all compact operators (Chapter 2); it does not, however, contain all bounded ones, and neither does even the C^{∗}-algebra generated by this set (Chapter 3). An unexpected fact is that these C^{∗}-algebras are always contained in certain C^{∗}- algebra A which does not depend on the underlying space (see Chapter 3 for a more precise statement). An attempt to construct a Toeplitz calculus on Bergman-type spaces, alluded to above, can be found in Chapter 4; it seems that the role of the Nagy-Foias dilation could be played by certain ”dilating” to the Calkin algebra. The Berezin transform, which plays an important role in the questions concerning compactness of Toeplitz operators (cf. [6], [2], [27]) seems to be important also from the viewpoint of the theory developed in Chapter 4; therefore the last Chapter 5 is devoted to its more detailed study.

Most methods are functional-analytic, or belong to operator theory and complex function theory. Rudiments of interpolation theory, potential theory and Riemannian geometry are needed in Chapter 5; algebraic topology (cohomotopy groups) appears in Remark 3.26.

RESULTS OF THE THESIS

Chapter 1 contains some preparatory material; most results are more or less known, although frequently their proofs cannot be found in the literature (”folk theorems”), so they are assembled here. The items 1.12, 1.15, and 1.18 are mine, and so are all presented proofs.

1See the end of Chapter 1 for terminology and notation.

In Chapter 2, density of Toeplitz operators on Bergman-type spaces is considered. It is proved that they are SOT-dense in the set of all bounded operators and that their norm closure contains all compact operators. A slightly simplified version of my results from [16] is reproduced here, together with a new proof and a generalization to arbitrary domains Ω ⊂ C^{N}.

In Chapter 3, it is proved that the C^{∗}-algebra generated by the Toeplitz operators does not contain all bounded linear operators. An unexpected fact is that this algebra is always contained in a certain C^{∗}-algebra which is the same (more precisely: spatially C^{∗}-isomorphic) for A^{2}(D), A^{2}(C), H^{2}, A^{2}(Ω) for a wide class of domains Ω ⊂ C, and only slightly different for A^{2}(C^{N}) (the difference seems to be of topological nature and related to the dimension, cf. Remark 3.26). A related class of spaces H^{2}(ρ) is also discussed.

In Chapter 4, an attempt is made to construct an analogue of the classical Toeplitz (or symbol) calculus on Bergman-type spaces (see the beginning of the chapter for a more explicit discussion). The problem is rather difficult, and only a partial solution is presented. Lemma 4.1 follows from the results of [3] or [23]; a very short and elegant proof is presented here, based on our Theorem 2.4. In 4.3 — 4.6, the techniques due to Calkin [10] are developed; 4.12 mentions some classical results, while 4.16 was inspired by Douglas ([12], 7.47 ff.). The proof of Lemma 4.19 is reproduced here from [6] (slightly simplified) for the purpose of subsequent discussion.

Some of the results from Chapters 1 and 4, as well as results of other authors ([5], [6], some preprints of Zhu, Stroethoff, Zheng, etc.) from adjacent areas, indicate that a fundamental role in the theory is played by the Berezin transform B; for this reason, the last chapter is devoted to its study. The operator B is shown to be bounded on various L^{p} spaces and to be related to the Laplace or the Laplace-Beltrami operator, respectively. For some classes of functions f, conditions for f to be invariant under B are given, and the iterates B^{n}f are shown to converge. The items 5.1 – 5.4 and 5.16 are more or less known (”folk theorems”), and so is perhaps 5.29; 5.18 and 5.28 are classical; 5.12 is a generalization of the Schur test, and 5.25 may be compared with a theorem in [19] (see Remark 5.26).

Unless stated otherwise, all the results to follow belong to the present author. Proofs are usually supplied for the ”folk theorems” mentioned above. When other authors’ results are needed, the references are always given in the text and, except for 4.3 – 4.6 and 4.19, the proofs are not reproduced.

ADDENDUM

After finishing this thesis, the author has received a preprint of Berger and Coburn [BC], and also a new article of Zhu [Z] has appeared. In [BC], our Theorem 2.4 is proved for the special case Ω = C^{N} (Theorem 9); the proof makes heavy use of the machinery developed by Berezin, but probably could be reduced to the method employed in our ”first proof” of Theorem 2.4. On page 38, they conjecture that ”The C^{∗}-algebra generated by all {T_{g} : g bounded } is evidently very large. Despite Theorem 16, this algebra could contain all bounded operators.” A negative answer to this conjecture is provided by our results in Chapter 3 (Theorem 3.27). In [Z], 5.6 and 5.7 appear, and 5.14 is reported to have been proved in the special case p = 2.

[BC] C.A. Berger, L.A. Coburn: Berezin-Toeplitz estimates. Preprint, ? October 1990.

[Z] K.H. Zhu: On certain unitary operators and composition operators. Proc. Symp. Pure Math. 51, 371 – 385. Providence, 1990.

Chapter 1. INTRODUCTION

The purpose of this chapter is to set up notation and to introduce some basic definitions. The style is brief; some proofs are omitted and can be found e.g. in [2] or [6]. The symbol C will allways denote the complex plane, D = {z ∈ C : |z| < 1} the open unit disc, T its boundary, the unit circle; Z is the set of all integers, and N the set of all nonnegative ones.

Convention: By a domain in C^{N} we mean an open, connected, nonempty and proper subset Ω of C^{N} (written: Ω ⊂ C^{N}), for some integer N ≥ 1. Thus, C^{N} itself is not considered to be a domain. We shall write Ω ⊆ C^{N} to express that Ω is either a domain in C^{N} in the above sense, or Ω = C^{N}.

The symbol dz denotes the Lebesgue measure in C^{N}, for all N ≥ 1, while L^{p}(Ω, dρ) stands for the usual Lebesgue space on Ω with respect to a measure ρ; if ρ is omitted, the Lebesgue measure is understood.

If Ω is a domain in C^{N} and 1 ≤ p ≤ ∞, we may define

A^{p}(Ω) := {f ∈ L^{p}(Ω, dz) : f is analytic on Ω}.

This is a closed subspace of L^{p}(Ω, dz). For p = ∞, A^{∞}(Ω) = H^{∞}(Ω), the space of all bounded analytic functions on Ω. For p = 2, A^{2}(Ω) becomes a Hilbert space (with the inner product from L^{2}). The spaces A^{2}(Ω) will be termed Bergman spaces ; the space A^{2}(D), which is of particular interest, will be referred to as the Bergman space. The latter will usually be considered as a subspace not of L^{2}(D, dz), but of L^{2}(D, dν), with the norm and scalar product modified accordingly; here

dν(z) = \frac{1}{π} dz

is a multiple of the Lebesgue measure chosen so that D had measure 1. If f = ∑_{n=0}^{∞} f_{n}z^{n} is holomorpic^{2} on

D, a simple calculation shows that

∫_{D} |f(x)|^{2} dν(x) = ∑_{n=0}^{∞} \frac{|f_{n}|^{2}}{n + 1} .

Consequently, f ∈ A^{2}(D) if and only if the last expression is finite. The scalar product of f and g = ∑_{n=0}^{∞} g_{n}z^{n},

f, g ∈ A^{2}(D), is given by

⟨f, g⟩_{A^{2} (D)} = ∑_{n=0}^{∞} \frac{f_{n}\overline{g_{n}}}{n + 1} .

The set

(1)  \{\sqrt{n + 1} z^{n}\}_{n∈N}

is an orthonormal basis for A^{2}(D). The polynomials are dense in A^{2}(D).

(Observe that the last sentence, in general, need not be valid for A^{2}(Ω) . If we take Ω = D \ ⟨0, 1), then the closure of the polynomials in A^{2}(Ω) is precisely A^{2}(D) ⊂ A^{2}(Ω) . The Riemann mapping function Φ : Ω → D belongs to A^{2}(Ω) – even to A^{∞}(Ω) – but not to A^{2}(D).)

The space A^{2}(D) is a reproducing kernel space in the sense of Aronszajn [1]. Denote P_{+} the orthogonal projection of L^{2}(Ω) onto A^{2}(Ω) .

Proposition 1.1. For each λ ∈ Ω ⊂ C^{N}, the linear functional f ↦ f(λ) on A^{2}(Ω) is bounded; consequently, f(λ) = ⟨f, g_{λ}⟩ > for some g_{λ} ∈ A^{2}(Ω) . Further,

∥g_{λ}∥ ≤ \frac{1}{(γ_{N}R^{2N})^{1/2}} ,

2a synonym for analytic throughout this paper where γ_{N} is the (Lebesgue) volume of the unit ball of C^{N} and R = dist(λ, C^{N} \ Ω).

Proof. Let R be the distance of λ to C^{N} \ Ω and

F_{λ}(z) = \{ \begin{matrix}0 & if |z − λ| ≥ R, \\ (γ_{N}R^{2N})^{−1} & if |z − λ| < R. \end{matrix}

Then

∫_{Ω} |F_{λ}(z)|^{2} dz = γ_{N}R^{2N} (\frac{1}{γ_{N}R^{2N}} )^{2} = \frac{1}{γ_{N}R^{2N}} < +∞,

so F_{λ} ∈ L^{2}(Ω). Hence, for arbitrary f ∈ A^{2}(Ω) ,

⟨f, P_{+} F_{λ}⟩_{A^{2} (Ω)} = ⟨f, F_{λ}⟩_{L^{2} (Ω)} = \frac{1}{γ_{N}R^{2N}} ∫_{|z−λ|<R} f(z) dz,

which equals f(λ) by the mean value theorem; hence, we may take g_{λ} = P_{+} F_{λ}. Finally,

∥g_{λ}∥_{2}^{2} ≤ ∥F_{λ}∥_{2}^{2} = \frac{1}{γ_{N}R^{2N}} .

□ For Ω = D, the reproducing kernel may be written down explicitly – namely,

g_{λ}(z) = \frac{1}{(1 − \overline{λ}z)^{2}} = ∑_{n=0}^{∞}(n + 1)\overline{λ}^{n}z^{n}.

Let Ω be a domain in C^{N} and ϕ ∈ L^{∞}(Ω) . We define the multiplication operator M_{ϕ} : L^{2}(Ω) → L^{2}(Ω), the Toeplitz operator T_{ϕ} : A^{2}(Ω) → A^{2}(Ω) and the Hankel operator H_{ϕ} : A^{2}(Ω) → L^{2}(Ω) ⊖ A^{2}(Ω) with symbol ϕ, respectively, by the formulas

M_{ϕ}f = ϕf,  T_{ϕ}f = P_{+} M_{ϕ}f,  H_{ϕ}f = (I − P_{+} )M_{ϕ}f.

These operators are clearly bounded, their norms not exceeding ∥ϕ∥_{∞}. If ϕ ∈ A^{∞}(Ω), H_{ϕ} is zero, while T_{ϕ} = M_{ϕ}↾A^{2}(Ω) . The mappings ϕ ↦ M_{ϕ}, ϕ ↦ T_{ϕ}, ϕ ↦ H_{ϕ} are linear. The formulas below are well-known in the theory of Toeplitz and Hankel operators on the Hardy space H^{2}; they remain in force in the present setting.

Proposition 1.2. For arbitrary f, g ∈ L^{∞}(Ω) ,

[T_{f}, T_{g}) := T_{fg} − T_{f}T_{g} = H∗ ; \overline{f}H_{g},

[T_{f}, T_{g}] := T_{f}T_{g} − T_{g}T_{f} = H∗ ; \overline{g}H_{f} − H∗ ; \overline{f}H_{g}.

In particular,

T_{fg} = T_{f}T_{g}, T\overline{g}T_{f} = T_{\overline{g}f}

when f ∈ L^{∞}(Ω) and g ∈ H^{∞}(Ω).

Proof.

T_{fg} − T_{f}T_{g} = P_{+} fg − P_{+} fP_{+} g = P_{+} f(I − P_{+} )g = H∗ ; \overline{f}H_{g};

[T_{f}, T_{g}] = [T_{g}, T_{f}) − [T_{f}, T_{g});

and H_{g} = 0 if g ∈ H^{∞}(Ω). □

The following proposition will prove handy when calculating some specific Toeplitz operators on A^{2}(D).

Proposition 1.3. Suppose f ∈ L^{∞}(D) is a radial function (i.e. f(z) depends only on |z| ). Then T_{f} is a diagonal operator with respect to the basis (1) with weights

c_{n}(F) := ∫_{0}^{1} F (t).(n + 1)t^{n} dt,

where F (t) := f(t^{1/2}), t ∈ ⟨0, 1).

Proof. Passing to the polar coordinates, we have

⟨T_{f}z^{n}, z^{m}⟩ = ∫_{D} f(z) z^{n}\overline{z}^{m} dν(z) = ∫_{0}^{1} \frac{1}{2π} ∫_{0}^{2π} f(r)r^{n+m} e^{(n−m)it} dt 2r dr.

If n ≠ m, this is zero; if n = m, it equals

∫_{0}^{1} F (r^{2})r^{2n} 2r dr = ∫_{0}^{1} F (t).t^{n} dt,

and the assertion follows immediately. □

The next proposition gives some feeling that the situation for A^{2}(D) is different from that for H^{2} – there are no nonzero compact Toeplitz operators on H^{2}.

Proposition 1.4. Assume that ϕ ∈ L^{∞}(Ω) and that the support suppϕ is a compact subset of Ω. Then M_{ϕ}↾A^{2}(D) is a compact operator; consequently, the operators T_{ϕ} and H_{ϕ} are also compact.

Proof. Suppose suppϕ = K is a compact subset of Ω, and set R = dist(K, C^{N} \ Ω) > 0. Assume that f_{n} ∈ A^{2}(Ω) is a sequence weakly converging to 0. Such sequence must be bounded – say, ∥f∥_{2} ≤ C ∀n. Then

|f_{n}(λ)| ≤ ∥f_{n}∥_{2}∥g_{λ}∥_{2} ≤ \frac{C}{\sqrt{γ_{N}R^{2N}}} ∀λ ∈ K,

whence

|ϕ(λ)f_{n}(λ)| ≤ \frac{C∥ϕ∥_{∞}}{\sqrt{γ_{N}R^{2N}}} ∀λ ∈ Ω.

Also, f_{n}→^{w}0 implies f_{n}(λ) = ⟨f_{n}, g_{λ}⟩ → 0 ∀λ ∈ Ω. Thus, we may apply the Lebesgue dominated convergence theorem to conclude that

∥ϕf_{n}∥_{2}^{2} = ∫_{Ω} |ϕ(λ)f_{n}(λ)|^{2} dλ = ∫_{K} |ϕ(λ)f_{n}(λ)|^{2} dλ → 0

as n → ∞. Hence M_{ϕ}↾A^{2}(Ω) maps weakly convergent sequences into norm convergent ones, and so must be compact. □

Define

V (D) := {f ∈ L^{∞}(D) : ess lim_{|z|↗1}f(z) = 0}.

Corollary 1.5. If ϕ ∈ V (D), then M_{ϕ}↾A^{2}(D), T_{ϕ} and H_{ϕ} are compact operators.

Proof. There exist ϕ_{n} ∈ V (D) such that suppϕ_{n} are compact subsets of D and ϕ_{n}⇉ϕ. Consequently, M_{ϕ_{n}} → M_{ϕ} in norm; since M_{ϕ_{n}} are all compact, so must be M_{ϕ}, and hence also T_{ϕ} and H_{ϕ}. □

If we take ϕ ∈ L^{∞}(Ω) , ϕ ≥ 0 whose support is a compact subset of Ω, then ⟨T_{ϕ}1, 1⟩ = ∫_{Ω} ϕ(z) dz > 0 and so T_{ϕ} ≠ 0; hence, indeed, T_{ϕ} is nonzero compact Toeplitz operator.

For Ω = C^{N}, the space A^{2}(C^{N}), defined as above, would consist only of constant zero; hence, we adopt a different definition in this case. For x, y ∈ C^{N}, write

\overline{x}y := ∑_{n=1}^{N} \overline{x_{n}}y_{n} and |x| := (\overline{x}x)^{1/2}.

(Thus, |x − y| is the usual Euclidean distance between x and y.) The Gaussian measure on C^{N} is, by definition,

dμ(z) = (2π)^{−N}e^{−|z|^{2} /2} dz.

Denote L^{p}(C^{N}, dμ) the usual Lebesgue spaces; L^{∞}(C^{N}, dμ) shall be occassionally abbreviated to L^{∞}(C^{N}), since they happen to coincide. Set, for 1 ≤ p ≤ ∞,

A^{p}(C^{N}) := {f ∈ L^{p}(C^{N}, dμ) : f is an entire function on C^{N}}.

Again, this is a closed subspace of L^{p}(C^{N}, dμ). A^{∞}(C^{N}) = H^{∞}(C^{N}), which contains only constant zero. For p = 2, A^{2}(C^{N}) is a Hilbert space, called the Fock or Segal-Bargmann space. Many results valid for A^{2}(D) carry over to the Fock space setting. For a multiindex n = (n_{1}, n_{2}, …, n_{N}) ∈ N^{N}, the following abbreviations will be employed:

a_{n} = a_{n_{1} ,n_{2} ,...,n_{N}} ,
z^{n} = z_{1}^{n_{1}}z_{2}^{n_{2}} … z_{N}^{n_{N}} (for z ∈ C^{N}),
n! = n_{1}!n_{2}! … n_{N}!, 2^{n} = 2^{n_{1} +n_{2} +...+n_{N}} .

If f is an entire function, f(z) = ∑_{n∈N^{N}} f_{n}z^{n}, then

∫_{C^{N}} |f(z)|^{2} dμ(z) = ∑_{n∈N^{N}} n!2^{n}|f_{n}|^{2}.

Consequently, f ∈ A^{2}(C^{N}) if and only if the last expression is finite. The inner product of f and g =

∑_{n∈N^{N}} g_{n}z^{n}, f, g ∈ A^{2}(C^{N}), is given by

⟨f, g⟩_{A^{2} (C^{N})} = ∑_{n∈N^{N}} n!2^{n}f_{n}\overline{g_{n}}.

The set

(2)  {(n!2^{n})^{−1/2} z^{n}}_{n∈N^{N}}

is an orthonormal basis of A^{2}(C^{N}). The polynomials are dense in A^{2}(C^{N}). Once again, A^{2}(C^{N}) is a reproducing kernel space; the reproducing kernel at λ ∈ C^{N} is given by

g_{λ}(z) = e^{\overline{λ}z/2},

and ∥g_{λ}∥_{2} = e^{|λ|^{2} /4}. For ϕ ∈ L^{∞}(C^{N}, dμ) = L^{∞}(C^{N}), the operators M_{ϕ}, T_{ϕ} and H_{ϕ} may be defined in the same way as for A^{2}(Ω) ; of course, P_{+} will be the orthogonal projection of L^{2}(C^{N}, dμ) onto A^{2}(C^{N}) this time. These operators are bounded, their norms not exceeding ∥ϕ∥_{∞}. The maps

ϕ ↦ M_{ϕ},  ϕ ↦ T_{ϕ},  ϕ ↦ H_{ϕ}

are linear. The following propositions are analogies of Propositions 1.2 — 1.5; their proofs are similar, and therefore omitted.

Proposition 1.6. For arbitrary f, g ∈ L^{∞}(C^{N}),

[T_{f}, T_{g}) := T_{fg} − T_{f}T_{g} = H∗ ; \overline{f}H_{g},

[T_{f}, T_{g}] := T_{f}T_{g} − T_{g}T_{f} = H∗ ; \overline{g}H_{f} − H∗ ; \overline{f}H_{g}.

In particular,

T_{fg} = T_{f}T_{g}, T\overline{g}T_{f} = T_{\overline{g}f}

when f ∈ L^{∞}(C^{N}) and g ∈ H^{∞}(C^{N}).

Proposition 1.7. Suppose f ∈ L^{∞}(C^{N}) is a radial function (i.e. f(z) depends only on |z_{1}|, |z_{2}|, …, |z_{n}|). Then T_{f} is a diagonal operator with respect to the basis (2) with weights

c_{n}(F) := ∫_{0}^{∞} ∫_{0}^{∞} … ∫_{0}^{∞} F (t)\frac{t^{n}e^{−(t_{1} +t_{2} +...+t_{N})}}{n!} dt,

where F (t) := f(\sqrt{2t_{1}}, \sqrt{2t_{2}}, …, \sqrt{2t_{N}}), t ∈ ⟨0, +∞)^{N}.

Proposition 1.8. Assume that ϕ ∈ L^{∞}(C^{N}) and suppϕ is compact. Then M_{ϕ}↾A^{2}(C^{N}), T_{ϕ} and H_{ϕ} are compact operators.

Corollary 1.9. If ϕ ∈ V (C^{N}), where

V (C^{N}) := {f ∈ L^{∞}(C^{N}) : ess lim_{|z|→+∞} f(z) = 0},

then M_{ϕ}↾A^{2}(C^{N}), T_{ϕ} and H_{ϕ} are compact operators.

The Corollaries 1.5 and 1.9 cannot be inverted — there exist functions ϕ ∈ L^{∞}(D) \ V (D) and ϕ ∈ L^{∞}(C^{N}) \ V (C^{N}), respectively, such that T_{ϕ} are compact operators. Furthermore, Toeplitz operators both on A^{2}(D) and A^{2}(C^{N}) may be well-defined and bounded also for some ϕ ∉ L^{∞}. It seems that boundedness or compactness of T_{ϕ} is not determined by the boundedness of ϕ or its vanishing near the boundary, respectively, but rather by these properties of the image ϕ̃ of ϕ under certain smoothing transformations. For the Fock space, this result is due to Berger and Coburn [6]. If ϕ ∈ L^{2}(C^{N}, dμ) and λ ∈ C^{N}, define k_{λ} := g_{λ}/∥g_{λ}∥, and

ϕ̃(λ) := ⟨ϕk_{λ}, k_{λ}⟩ = ∫_{C^{N}} ϕ(z) e^{\frac{\overline{λ}z}{2}+ \frac{λ\overline{z}}{2}− \frac{|λ|^{2}}{2}} dμ(z) =

= ∫_{C^{N}} ϕ(z) e^{−|λ−z|^{2} /2} \frac{dz}{(2π)^{N}} .

ϕ̃ is called the Berezin transform of ϕ; the definition may be extended to ϕ ∈ L^{1}(C^{N}, dμ) in an obvious way.

Theorem 1.10. Let ϕ ∈ L^{2}(C^{N}, dμ).

(i) If T_{ϕ} is bounded, ϕ̃ ∈ L^{∞}(C^{N}); if it is compact, ϕ̃ ∈ V (C^{N}).

(ii) M_{ϕ}↾A^{2}(C^{N}) is bounded iff (|ϕ|^{2}) ∈ L^{∞}(C^{N}), and compact iff (|ϕ|^{2}) ∈ V (C^{N}).

(iii) If ϕ ≥ 0, then T_{ϕ} is bounded iff ϕ̃ ∈ L^{∞}(C^{N}), and compact iff ϕ ∈ V (C^{N}).

For the Bergman space A^{2}(D), the best reference seems to be Axler [2]. The Möbius transformation ω_{λ} corresponding to λ ∈ D is, by definition, the function

x ↦ ω_{λ}(x) := \frac{x − λ}{1 − \overline{λ}x},

which maps D bijectively onto itself. The function

d(x, y) := |ω_{x}(y)|

is called the pseudo-hyperbolic metric; it is, indeed, a metric on D. Denote D_{h}(λ, R) the disc with center λ and radius R with respect to this metric. If ϕ ∈ L^{1}(D, dν), denote

ϕ^{♭}(λ) := \frac{1}{ν(D_{h}(λ, 1/2))} ∫_{D_{h}(λ,1/2)} ϕ(x) dν(x),

the mean value of ϕ on D_{h}(λ, 1/2).

Theorem 1.11. Let ϕ ∈ L^{2}(D, dν).

(a) M_{ϕ}↾A^{2}(D) is bounded iff (|ϕ|^{2})^{♭} ∈ L^{∞}(D), and compact iff (|ϕ|^{2})^{♭} ∈ V (D).

(b) If ϕ ≥ 0, T_{ϕ} is bounded iff ϕ^{♭} ∈ L^{∞}(D), and compact iff ϕ^{♭} ∈ V (D). The number 1/2 may be replaced by arbitrary R ∈ (0, 1).

The Berezin transform may be defined on D, too. For ϕ ∈ L^{2}(D, dν) and λ ∈ D, set k_{λ} := g_{λ}/∥g_{λ}∥ and

ϕ̃(λ) := ⟨ϕk_{λ}, k_{λ}⟩ = ∫_{D} ϕ(z) . \frac{(1 − |λ|^{2})^{2}}{|1 − \overline{λ}z|^{4}} dν(z).

Using the last theorem, it is possible to prove a complete analogue of Theorem 1.10. Theorem 1.12. Let ϕ ∈ L^{2}(D).

(i) If T_{ϕ} is bounded, ϕ̃ ∈ L^{∞}(D); if it is compact, ϕ̃ ∈ V (D). (ii) M_{ϕ}↾A^{2}(D) is bounded iff (|ϕ|^{2}) ∈ L^{∞}(D), and compact iff (|ϕ|^{2}) ∈ V (D). (iii) If ϕ ≥ 0, then T_{ϕ} is bounded iff ϕ̃ ∈ L^{∞}(D), and compact iff ϕ ∈ V (D). Lemma 1.13. When |λ| → 1, k_{λ}→^{w}0. Proof. Since ∥k_{λ}∥ = 1 ∀λ ∈ D and the polynomials are dense in A^{2}(D), it is enough to check that

⟨p, k_{λ}⟩ → 0 as |λ| → 1 when p is a polynomial; but that’s immediate, because

⟨z^{n}, k_{λ}⟩ = (1 − |λ|^{2})⟨z^{n}, g_{λ}⟩ = (1 − |λ|^{2}) . λ^{n} → 0 as n → +∞.

□ Remark 1.14. The last lemma remains in force for A^{2}(C^{N}), too; the proof is similar. □ Lemma 1.15. For each R ∈ (0, 1), there is C = C(R) > 0 such that

∫_{D} ϕ(z) \frac{(1 − |λ|^{2})^{2}}{|1 − \overline{λ}z|^{4}} dν(z) ≥ \frac{C^{2}}{ν(D_{h}(λ, R))} ∫_{D_{h}(λ,R)} ϕ(z) dν(z)

for every nonnegative function ϕ on D.

Proof. It suffices to find C > 0 so that

\frac{(1 − |λ|^{2})^{2}}{|1 − \overline{λ}z|^{4}} ≥ \frac{C^{2}}{ν(D_{h}(λ, R))} ∀z ∈ D_{h}(λ, R).

This is equivalent to

\frac{1}{|1 − \overline{λ}z|^{2}} ≥ C. \frac{1 − R^{2}|λ|^{2}}{R(1 − |λ|^{2})^{2}} ∀z ∈ D_{h}(λ, R).

But, if z ∈ D_{h}(λ, R),

|1 − \overline{λ}z| = |λ| |\frac{1}{\overline{λ}} − z| ≤ |λ| |\frac{1}{λ} − \frac{|λ| − R}{1 − |λ|R}| = \frac{1 − |λ|^{2}}{1 − R|λ|};

hence it is enough to manage that

(\frac{1 − R|λ|}{1 − |λ|^{2}})^{2} ≥ C.\frac{1 − R^{2}|λ|^{2}}{R(1 − |λ|^{2})^{2}} ,

and a short computation reveals that C = R(1 − R)/(1 + R) will do. □

Proof. (of Theorem 1.12) (i) If T_{ϕ} is bounded,

|ϕ̃(λ)| = |⟨ϕk_{λ}, k_{λ}⟩| = |⟨T_{ϕ}k_{λ}, k_{λ}⟩| ≤ ∥T_{ϕ}∥.∥k_{λ}∥^{2} = ∥T_{ϕ}∥.

If T_{ϕ} is compact, ∥T_{ϕ}k_{λ}∥ → 0 as |λ| → 1 owing to Lemma 1.13; hence ϕ̃(λ) = ⟨T_{ϕ}k_{λ}, k_{λ}⟩ → 0 as |λ| → 1. (ii) M_{ϕ}↾A^{2}(D) is bounded or compact, respectively, if and only if

(M_{ϕ}↾A^{2}(D))^{∗}(M_{ϕ}↾A^{2}(D)) = T_{|ϕ|^{2}}

is; hence, (ii) reduces to (iii).

(iii) The ”only if” part is contained in (i). To prove the ”if” part, suppose that ϕ ≥ 0 and ϕ̃ ∈ L^{∞}(D); then, owing to Lemma 1.15, ϕ^{♭} ∈ L^{∞}(D), and so T_{ϕ} is bounded by Theorem 1.11(b). The statement concerning compactness may be proved in a similar manner. □

It is actually possible to go a little further and define the Berezin transform for linear operators on A^{2}(C^{N}) or A^{2}(D) by the formula

T̃(λ) := ⟨T k_{λ}, k_{λ}⟩; k_{λ} = \frac{g_{λ}}{∥g_{λ}∥}, λ ∈ C^{N} or D.

The operator T need not be bounded, and it suffices if its domain of definition contains all k_{λ}, λ ∈ C^{N} or D.

Proposition 1.16. (i) If T is bounded, T̃ ∈ L^{∞} and ∥T̃∥_{∞} ≤ ∥T ∥. If T is compact, T̃ ∈ V.

(ii) If T = T_{ϕ} is a Toeplitz operator, ϕ ∈ L^{2}, then T̃_{ϕ} = ϕ̃.

(iii) If T is diagonal with respect to the basis (1) or (2), respectively, then T̃ is radial. Here L^{∞} is either L^{∞}(D) or L^{∞}(C^{N}), and similarly for L^{2} and V.

Proof. (i) |T̃(λ)| ≤ ∥T ∥.∥k_{λ}∥^{2} = ∥T ∥; if T is compact, T k_{λ} → 0 as |λ| ↗ 1 owing to Lemma 1.13, and so T̃(λ) → 0 as |λ| ↗ 1. The proof for the Fock space is similar (cf. Remark 1.14).

(ii) Immediate.

(iii) We will deal with A^{2}(D) only, the proof of the other case being similar. Suppose T is diagonal with weights c_{n}. Then

T g_{λ} = ∑_{n=0}^{∞}(n + 1)\overline{λ}^{n}c_{n}z^{n},

whence

T̃(λ) = (1 − |λ|^{2})^{2}. ∑_{n=0}^{∞}(n + 1)c_{n}|λ|^{2n},

and that’s a radial function. □

Corollary 1.17. If ϕ ∈ L^{2}(C^{N}) or L^{2}(D) is radial, then so is ϕ̃.

Proof. Combine Propositions 1.3 and 1.7 with Proposition 1.16(iii). □

Remark 1.18. The item (i) of the last proposition bears certain resemblance to the same items of Theorems 1.10 and 1.12. It should be noted, however, that this analogy fails for (iii): there exist unbounded positive operators T such that T̃ ∈ L^{∞}, and non-compact positive operators T such that T̃ ∈ V. For completeness, we present below an example of the latter (on A^{2}(D)).

Example 1.19. Let A be the operator on A^{2}(D) sending z^{n} into c_{n}z^{n} (n ∈ N), where

c_{n} = \{ \begin{matrix}1 & if n = 2^{k} − 1, k = 1, 2, …, \\ 0 & otherwise. \end{matrix}

Then A is a bounded positive noncompact linear operator; we are going to show that Ã ∈ V (D). From the proof of Proposition 1.16, we see that

Ã(λ) = (1 − |λ|^{2})^{2}. ∑_{k=1}^{∞}2^{k}|λ|^{2(2^{k}+1)} = R(1 − R)^{2} ∑_{k=1}^{∞}2^{k}R^{2^{k}},

where, for brevity, R = |λ^{2}|. The function x ↦ 2^{x}R^{2^{x}} is increasing on the interval (0, x_{R}) and decreasing on (x_{R}, +∞), where x_{R} is given by 2^{x_{R}} = −\frac{1}{ln R}. Let k_{R} be the least integer greater or equal to x_{R}. Then

∑_{k=1}^{∞}2^{k}R^{2^{k}} = ∑_{k=1}^{k_{R}}2^{k}R^{2^{k}} + ∑_{k=k_{R}+1}^{∞} 2^{k}R^{2^{k}} ≤ ∑_{k=1}^{k_{R}}2^{k} + ∫_{x_{R}}^{∞} 2^{x}R^{2^{x}} dx = 2^{k_{R}+1} + [\frac{R^{2^{x}}}{lnR ln2}]_{x_{R}}^{∞} =

= 2^{k_{R}+1} + \frac{1/e}{− lnR ln2} ≤ 4\frac{1}{− lnR} + \frac{1/e}{− ln2 lnR} ≤ \frac{5}{− lnR}.

Consequently,

0 ≤ Ã(λ) ≤ R(1 − R)^{2} \frac{5}{− lnR} = O(1 − R) as R ↗ 1,

which proves that Ã ∈ V (D). □

Notation. If H is a Hilbert space, B(H) stands for the C^{∗}-algebra of all bounded linear operators on H (with the operator norm ∥·∥), Comp (H) for the space of all compact operators from B(H); Comp (H) will be abbreviated Compwhen it’s clear what H is. The norm is denoted ∥ · ∥ on all spaces; ∥ · ∥_{p} is sometimes used for the norm in L^{p} or A^{p}. Where ambiguity might arise, the space where the norm is taken is given as an index, e.g. ∥f∥_{L^{2} (D,dν)} . Similar conventions will be observed for scalar products ⟨ . , . ⟩. SOT and WOT are abbreviations for the strong and weak operator topologies on B(H), respectively. If M is a subset of B(H), closM denotes the closure of M in the operator norm topology. When discussing B(A^{2}(D)) or B(A^{2}(C^{N})), diag(c_{n}) means the diagonal operator with weights c_{n} with respect to the basis (1) or (2), respectively. The symbol ⇉ is employed to denote uniform convergence; →^{w} means weak convergence in a Banach space. A bar over a subset of C^{N} denotes the closure; a bar over a number, a function, etc., the complex conjugate. For other symbols, not mentioned in this Introduction, consult the List of Notation.

By a Bergman-type space, we mean either a Bergman space A^{2}(Ω) , Ω ⊂ C^{N}, or the Fock space A^{2}(C^{N}).

Chapter 2. DENSITY OF TOEPLITZ OPERATORS

Let T_{ϕ} be a Toeplitz operator on the Hardy space H^{2}. It is easily seen that

T_{z}^{∗}T_{ϕ}T_{z} = T_{ϕ} for any ϕ ∈ L^{∞}(T).

According to a classical result, the converse also holds: if some operator T : H^{2} ⟶ H^{2} satisfies T_{z}^{∗}T T_{z} = T, then T = T_{ϕ} for some ϕ ∈ L^{∞}(T). This result serves as a starting point for the theory of symbols of operators (cf. [24]). It shows that, loosely speaking, there are only few Toeplitz operators on H^{2}.

In my paper [15], I have shown that Toeplitz operators on A^{2}(D) don’t admit the characterization as above. More precisely, if AT_{ϕ}B = T_{ϕ} for all ϕ ∈ L^{∞}(D), then A = cI, B = c^{−1}I for some (nonzero) complex number c. The proof works also for A^{2}(Ω) and A^{2}(C^{N}).

A natural question to ask is if this is not because there are, loosely speaking, more operators which are Toeplitz than in the classical (i.e. H^{2}) case. To put it precisely, we can ask if following statements are true:

(1^{o}) the Toeplitz operators are dense (in some topology) in B(A^{2}(D));

(2^{o}) every finite dimensional operator is Toeplitz;

(3^{o}) for any linearly independent elements f, g ∈ A^{2}(D), there exists ϕ ∈ L^{∞}(D) such that T_{ϕ}f = g. Clearly (2^{o}) implies (1^{o}) in the strong operator topology, and either (1^{o}) or (3^{o}) implies the impossibility of the above-mentioned characterization AT_{ϕ}B = T_{ϕ}.

The statement (2^{o}) is easily seen not to be true. For example, there is no ϕ ∈ L^{∞}(D) such that T_{ϕ} = ⟨. , 1⟩1. This fact is an easy consequence of the Müntz-Szász theorem for L^{2} spaces (see, for instance, [14]). In fact, this theorem yields a much stronger result: if T_{ϕ} = ⟨. , f⟩g for some ϕ ∈ L^{∞}(D) and f, g ∈ A^{2}(D) , f(z) = ∑_{0}^{∞} f_{n}z^{n} , g(z) = ∑_{0}^{∞} g_{n}z^{n} , then

∑_{f_{n}=0} \frac{1}{n} < +∞ and ∑_{g_{n}=0} \frac{1}{n} < +∞ .

(Loosely speaking, only few Taylor coefficients of f and g can be zero.) It’s a conjecture of author’s that in fact there are no finite-dimensional Toeplitz operators in B(A^{2}(D)) at all.

(3^{o}) is readily seen to be false, too. It suffices to take f = 1: if there were, for every g ∈ A^{2}(D), some ϕ ∈ L^{∞}(D) such that g = T_{ϕ}1 (= P_{+} ϕ), then the mapping (here (A^{2}(D))^{d} stands for the dual space of A^{2}(D))

A : L^{∞}(D) ⟶ (A^{2}(D))^{d} , ϕ ⟼ ⟨. , P_{+} \overline{ϕ}⟩ = ⟨. , \overline{ϕ}⟩_{L^{2} (D)}

would be onto. Let B be the operator of inclusion of A^{2}(D) into L^{1}(D):

B : A^{2}(D) ⟶ L^{1}(D) , ψ ⟼ ψ .

This is a continuous operator (by the Schwarz inequality) and has A as its adjoint, B^{d} = A. By the Hausdorff normal solvability theorem (cf. [26], chapter VII, §5), Ran A is closed if, and only if, Ran B is closed. Because B is injective, Ran B is closed if and only if B is bounded below (just use the open mapping theorem). But the norm of z^{n} in L^{1}(D) is

∫_{D} |z^{n}| dz = ∫_{0}^{1} r^{n}.2r dr = \frac{2}{n + 2} ,

whereas the norm of z^{n} in A^{2}(D) is

∥z^{n}∥_{2} = (∫_{D} |z^{n}|^{2} dz)^{1/2} = (n + 1)^{−1/2} .

Consequently, B is not bounded below, so Ran A is not closed, and A cannot be surjective. (The last part of the argument can be avoided by evoking directly the fact that the closure of A^{2}(D) in L^{1}(D) is A^{1}(D), the space of all integrable analytic functions on D. Our method is more elementary. )

All the same, there is a weakened version of (3^{o}) that does hold and which, moreover, implies that (1^{o}) holds in SOT.

Theorem 2.1. Let^{3} Ω ⊆ C^{N}, T ∈ B(A^{2}(Ω) ), F_{i}, G_{i} ∈ A^{2}(Ω) (i = 1, 2, …, N). Then there exists ϕ ∈ L^{∞}(Ω) such that

⟨T_{ϕ}F_{i}, G_{i}⟩ = ⟨T F_{i}, G_{i}⟩ , i = 1, 2, …, N.

Proof. We shall prove the theorem for the case Ω ≠ C^{N}; the proof for the Fock space is perfectly similar. Let f_{1}, f_{2}, …, f_{n}, resp. g_{1}, g_{2}, …, g_{m} be a basis of the (finite-dimensional) subspace of A^{2}(Ω) generated by F_{1}, …, F_{N}, resp. G_{1}, …, G_{N}. Clearly it’s sufficient to find ϕ ∈ L^{∞}(Ω) such that

⟨T_{ϕ}f_{i}, g_{j}⟩ = ⟨T f_{i}, g_{j}⟩ for all i = 1, …, n and j = 1, …, m.

Consider the operator R : L^{∞}(Ω) ⟶ C^{n×m}, defined by the formula

(Rϕ)_{ij} = ∫_{Ω} ϕ(z)f_{i}(z)\overline{g_{j}(z)}dz = ⟨T_{ϕ}f_{i}, g_{j}⟩.

Suppose some u ∈ C^{n×m} is orthogonal to the range of R, i.e.

∑_{i=1}^{n} ∑_{j=1}^{m}(Rϕ)_{ij}\overline{u}_{ij} = 0 for all ϕ ∈ L^{∞}(Ω) .

This means that

(3)  ∫_{Ω} ϕ(z). ∑_{i=1}^{n} ∑_{j=1}^{m} \overline{u}_{ij}f_{i}(z)\overline{g_{j}(z)}dz = 0  for all ϕ ∈ L^{∞}(Ω) ,

which implies

(4)  ∑_{i=1}^{n} ∑_{j=1}^{m} \overline{u}_{ij}f_{i}(z)\overline{g_{j}(z)} = 0

dz-almost everywhere in Ω; since the left-hand side is obviously continuous on Ω, this equality holds, in fact, on the whole of Ω. Thus, the function

F (x, y) = ∑_{i=1}^{n} ∑_{j=1}^{m} \overline{u}_{ij}f_{i}(x)\overline{g_{j}(\overline{y})},

which is analytic in Ω × Ω, equals zero whenever x = \overline{y}. By a well-known uniqueness theorem, this implies that F is identically zero on Ω × Ω. Because the functions f_{i}, i = 1, 2, …, n, are linearly independent, we have

∑_{j=1}^{m}u_{ij}g_{j}(\overline{y}) = 0 for all y ∈ Ω, i = 1, 2, …, n;

but g_{j}, j = 1, 2, …, m, are also linearly independent, and so u_{ij} = 0 for all i, j, i.e. u = 0. This means that the range of R is all of C^{n×m}, which immediately yields the desired conclusion. □

Corollary 2.2. The set T = {T_{ϕ} : ϕ ∈ L^{∞}(Ω) } is dense in B(A^{2}(Ω) ) in SOT (the strong operator topology). The assertion holds also for Ω = C^{N}.

3Remember that, according to the convention from the Introduction, this means that either Ω is a domain in C^{N}, or that Ω = C^{N}.

Proof. In view of the preceding theorem, it is certainly dense in WOT (the weak operator topology); and because T is a subspace, i.e. a convex set, its WOT- and SOT- closures coincide. □

Note that the crucial step in the proof of Theorem 2.1 was the implication (3)⇒(4). Thus, the theorem remains in force if we have (3) only for ϕ ∈ D(Ω) (the set of all infinitely differentiable functions on Ω whose support is a compact subset of Ω) — any w^{∗}-dense subset of L^{∞}(Ω) will do. Consequently, the following theorem holds.

Theorem 2.3. Let Ω ⊆ C^{N}. Then the set T_{1} = {T_{ϕ} : ϕ ∈ D(Ω)} is SOT-dense in B(A^{2}(Ω) ).

A natural question arises at this point — namely, whether the Toeplitz operators are not actually normdense in B(A^{2}(Ω) ). We shall see later that this is not the case — even the C^{∗}-algebra generated by them is smaller than B(A^{2}(Ω) ). It is true, however, that the norm closure of the Toeplitz operators contains all compact operators.

Theorem 2.4. Suppose Ω ⊆ C^{N}. Let T_{1} = {T_{ϕ} : ϕ ∈ D(Ω)}. Then closT_{1} = Comp (A^{2}(Ω) ).

Since T_{ϕ} ∈ Compfor ϕ ∈ D(Ω) in view of Propositions 1.4 and 1.8, the inclusion closT_{1} ⊂ Comp (A^{2}(Ω) ) is obvious; it remains to show that

Comp (A^{2}(Ω) ) ⊂ closT_{1}.

We are going to present two proofs. The first is shorter and works for arbitrary Ω ⊆ C^{N}; however, it is not constructive. The second applies only for D or C^{N}, involves much computation, but exhibits explicitely how to manage the approximation.

Proof. First (short) proof. We begin by a simple lemma.

Lemma 2.5. Let Ω ⊆ C^{N}. Then the reproducing kernels g_{λ}, λ ∈ Ω, span A^{2}(Ω) .

Proof. Suppose f ∈ A^{2}(Ω) is orthogonal to all g_{λ}, λ ∈ Ω. Then f(λ) = ⟨f, g_{λ}⟩ = 0 ∀λ ∈ Ω, i.e. f = 0. □

Recall that the dual of Comp (H), where H is a separable Hilbert space, may be identified with Trace(H), the space of all trace class operators on H equipped with the trace norm ∥ · ∥_{Tr}; the pairing is given by (K, T) ↦ Tr(KT) = Tr(T K), Tr being the trace.

Suppose that Ω ⊆ C^{N} and that closT_{1} is a proper subset of Comp (A^{2}(Ω) ). By the Hahn-Banach theorem, there exists T ∈ Trace(A^{2}(Ω) ), T ≠ 0, such that

Tr(T T_{ϕ}) = 0 ∀ϕ ∈ D(Ω).

Let A, B be two Hilbert-Schmidt operators such that T = AB^{∗}, {e_{n}}_{n=0}^{∞} an orthonormal basis for A^{2}(Ω) ,

f_{n} = Ae_{n}, g_{n} = Be_{n}. Then Tr(T T_{ϕ}) = Tr(B^{∗}T_{ϕ}A) = ∑_{n=0}^{∞} ⟨B^{∗}T_{ϕ}Ae_{n}, e_{n}⟩, and so the last condition may be

rewritten as

∑_{n=0}^{∞}⟨T_{ϕ}f_{n}, g_{n}⟩ = 0 ∀ϕ ∈ D(Ω),

or

∑_{n=0}^{∞}∫_{Ω} ϕ(λ)f_{n}(λ)\overline{g_{n}(λ)}dν(λ) = 0 ∀ϕ ∈ D(Ω)

(if Ω = C^{N}, replace dν by dμ). Because ϕ has compact support, ∑ ∥f_{n}∥^{2} < +∞ and ∑ ∥g_{n}∥^{2} < +∞, we may interchange the integration and summation signs, which yields

∫_{Ω} ϕ(λ)F (λ, \overline{λ}) dν(λ) (or dμ(λ)) = 0 ∀ϕ ∈ D(Ω),

where

F (x, y) = ∑_{n=0}^{∞}f_{n}(x)\overline{g_{n}(\overline{y})} = Tr(T G_{x,y}),

G_{x,y} = ⟨· , g_{x}⟩g\overline{y}.

It follows that F (λ, \overline{λ}) = 0 for almost all λ ∈ Ω; in other words, the function F (x, y), analytic on Ω × Ω, vanishes when x = \overline{y}. Appealing to the aforementioned uniqueness principle, F = 0 everywhere on Ω × Ω, i.e. Tr(T G_{x,y}) = 0 ∀x, y ∈ Ω. In view of the last lemma, Tr(T K) = 0 for all rank one operators K; by linearity and continuity, Tr(T K) = 0 for all compact K, whence T = 0 — a contradiction. The proof is complete. □

The other proof is a slightly simplified version of the author’s original one from [16]. It is based on two lemmas. We state and prove them first for A^{2}(D); the (slightly easier) case of A^{2}(C^{N}) will be discussed afterwards.

For m, n ∈ N and a ∈ D, let T_{(m,n,a)} be the operator on A^{2}(D) given by

f ⟼ ⟨f, g_{m,a}⟩g_{n,a} ,

where a ∈ D, m and n are non-negative integers, and g_{m,a} ∈ A^{2}(D) is given by the formula

g_{m,a}(x) := \frac{(m + 1)!z^{m}}{(1 − \overline{a}z)^{m+2}} .

One has

⟨f, g_{m,a}⟩ = f^{(m)}(a),

the m-th derivative of f ∈ A^{2}(D) at a ∈ D; thus,

⟨T_{(m,n,a)}f, g⟩ = f^{(m)}(a)\overline{g^{(n)}(a)}

for arbitrary f, g ∈ A^{2}(D).

Lemma 2.6. Let M, N be non-negative integers, a ∈ D, and denote

R_{(M,N,a,t)} = \frac{T_{(M,N,a+t)} − T_{(M,N,a)}}{2t} − i\frac{T_{(M,N,a+it)} − T_{(M,N,a)}}{2t} .

Then R_{(M,N,a,t)} tends to T_{(M+1,N,a)} (in norm) as the real number t tends to zero:

lim_{R∋t→0} ∥R_{(M,N,a,t)} − T_{(M+1,N,a)} ∥ = 0 .

Similarly,

lim_{R∋t→0} ∥R′_{(M,N,a,t)} − T_{(M,N+1,a)} ∥ = 0 ,

where

R′_{(M,N,a,t)} = \frac{T_{(M,N,a+t)} − T_{(M,N,a)}}{2t} + i\frac{T_{(M,N,a+it)} − T_{(M,N,a)}}{2t} .

(Here i stands for \sqrt{−1}.)

Proof. Let F, G ∈ A^{2}(D) and denote, for a while, f = F^{(M)} and g = G^{(N)}. Then
⟨(R_{(M,N,a,t)} − T_{(M+1,N,a)} ) F, G⟩ =

= \frac{f(a + t)\overline{g(a + t)} − f(a)\overline{g(a)}}{2t} − i\frac{f(a + it)\overline{g(a + it)} − f(a)\overline{g(a)}}{2t} − f′(a)\overline{g(a)}. (5)

Let

(6)  f(x) = ∑_{0}^{∞}f_{n}.(x − a)^{n} , g(x) = ∑_{0}^{∞}g_{n}.(x − a)^{n}

be the Taylor expansion of f, resp. g at a. These series are locally uniformly convergent in the disc |x − a| < 1 − a. Consequently, for |t| < 1 − |a| the right-hand side of (5) equals to

\frac{1}{2t} ( ∑_{\begin{matrix}m,n≥0 \\ (m,n)≠(0,0) \end{matrix}} f_{m}t^{m}.\overline{g}_{n}t^{n}) − \frac{i}{2t} ( ∑_{\begin{matrix}m,n≥0 \\ (m,n)≠(0,0) \end{matrix}} f_{m}.(it)^{m}.\overline{g}_{n}.(−it)^{n}) − f_{1} \overline{g}_{0} .

Rearranging all terms into one series, the terms corresponding to (m, n) = (1, 0) and (0, 1) cancel, and we get

(7)  \frac{1}{2} . ∑_{\begin{matrix}m,n≥0 \\ m+n≥2 \end{matrix}} f_{m}\overline{g}_{n}t^{m+n−1} (1 − i^{m−n+1} ) .

Let F_{n}, resp. G_{n} be the coefficients of the Taylor expansion of the function F, resp. G at a. Because f = F^{(M)}, we have

f_{m} = \frac{1}{m!}f^{(m)}(a) = \frac{1}{m!}F^{(M+m)} (a) = \frac{(M + m)!}{m!} F_{m+M} ,

and similarly for g_{n}. It follows that (7) is equal to

(8)  \frac{1}{2} . ∑_{\begin{matrix}m,n≥0 \\ m+n≥2 \end{matrix}} \frac{(M + m)!}{m!} F_{m+M} .\frac{(N + n)!}{n!} \overline{G}_{n+N} .t^{m+n−1} .(1 − i^{m−n+1} ) .

We are going to estimate the absolute value of the last expression in terms of ∥F ∥_{2} and ∥G∥_{2}. One has

∥F ∥_{2}^{2} = ∫_{D} |F (z)|^{2} dz ≥ ∫_{|z−a|<1−|a|} ∥F (z)∥^{2} dz =

= ∫_{0}^{1−|a|} ∫_{0}^{2π} ∑_{j,k≥0} F_{j}\overline{F}_{k}r^{j+k} e^{(j−k)it}.\frac{r}{π} dt dr ,

(we have passed to polar coordinates). Since the Taylor series

F (z) = ∑_{0}^{∞}F_{n}.(z − a)^{n}

is locally uniformly convergent on the disc |z−a| < 1−|a|, we can interchange the integration and summation signs and get

∥F ∥_{2}^{2} ≥ ∑_{j,k=0}^{∞} ∫_{0}^{1−|a|} ∫_{0}^{2π} F_{j}\overline{F}_{k}r^{j+k} e^{(j−k)it}.\frac{r}{π} dr dt =

= ∑_{k=0}^{∞} \frac{(1 − |a|)^{2k+2}}{k + 1} |F_{k}|^{2} . (9)

Similar argument holds for G. Denote, for a while,

α_{k} = \frac{(1 − |a|)^{k+1}}{(k + 1)^{1/2}} |F_{k}| , β_{k} = \frac{(1 − |a|)^{k+1}}{(k + 1)^{1/2}} |G_{k}| .

According to (9), α and β belong to l^{2} and

∥α∥_{2} ≤ ∥F ∥_{2} , ∥β∥_{2} ≤ ∥G∥_{2} .

Returning to our previous calculations, we see that the absolute value of (8) is not greater then

\frac{1}{2} . ∑_{\begin{matrix}m,n≥0 \\ m+n≥2 \end{matrix}} \frac{(M + m)!}{m!} |F_{m+M} | . \frac{(N + n)!}{n!} |G_{n+N} | . t^{m+n−1} . 2 =

= ∑_{\begin{matrix}m,n≥0 \\ m+n≥2 \end{matrix}} \frac{(M + m)!}{m!} \frac{(M + m + 1)^{1/2}t^{m− \frac{1}{2}}}{(1 − |a|)^{M+m+1}} α_{M+m} . \frac{(N + n)!}{n!} \frac{(N + n + 1)^{1/2}t^{n− \frac{1}{2}}}{(1 − |a|)^{N+n+1}} β_{N+n} .

Break this sum into three parts, namely,

∑_{m,n≥1} + ∑_{\begin{matrix}m=0 \\ n≥2 \end{matrix}} + ∑_{\begin{matrix}m≥2 \\ n=0 \end{matrix}} := σ_{1} + σ_{2} + σ_{3} .

Let’s first consider σ_{1}. Obviously

σ_{1} = ( ∑_{m=1}^{∞} \frac{(M + m)!}{m!} \frac{(M + m + 1)^{\frac{1}{2}} t^{m− \frac{1}{2}}}{(1 − |a|)^{M+m+1}} α_{M+m} ) ×

× (∑_{n=1}^{∞} \frac{(N + n)!}{n!} \frac{(N + n + 1)^{\frac{1}{2}} t^{n− \frac{1}{2}}}{(1 − |a|)^{N+n+1}} β_{N+n} ) . (10)

According to the Cauchy-Schwarz inequality, the first factor on the right-hand side is less than or equal to

∥α∥_{2} . ( ∑_{m=1}^{∞} \frac{(M + m)!^{2}}{m!^{2}} . \frac{(M + m + 1).t^{2m−1}}{(1 − |a|)^{2M+2m+2}} )^{1/2} = (11)

= ∥α∥_{2} . (t . ∑_{m=0}^{∞} \frac{(M + m + 1)!^{2}}{(m + 1)!^{2}} . \frac{(M + m + 2).t^{2m}}{(1 − |a|)^{2M+2m+4}} )^{1/2} =

∥α∥_{2} . t^{1/2}c_{1}(M, a, t),

where c_{1}(M, a, t) tends to a finite limit \frac{(M + 1)!^{2}(M + 2)}{(1 − |a|)^{2M+4}} as t^{2} → 0+, i.e. as t → 0.

A similar estimate holds for the second factor in (10). Putting the two estimates together, we see that

σ_{1} ≤ ∥α∥_{2} . ∥β∥_{2} . t . c_{2}(M, N, a, t)

≤ ∥F ∥_{2} . ∥G∥_{2} . t . c_{2}(M, N, a, t) , (12) where

c_{2}(M, N, a, t) = c_{1}(M, a, t)c_{1}(N, a, t)

tends to a finite limit as t → 0.

Now let’s turn our attention to σ_{2}. We have

σ_{2} ≤ \frac{M!(M + 1)^{1/2}}{(1 − |a|)^{M+1}} . ∥α∥_{2} . [∑_{n=2}^{∞} \frac{(N + n)!}{n!} \frac{(N + n + 1)^{1/2}}{(1 − |a|)^{N+n+1}} . t^{n−1} . β_{N+n} ] .

Using Cauchy-Schwarz inequality shows that the bracketed term is not greater then

(13)  ∥β∥_{2} . (∑_{n=2}^{∞} \frac{(N + n)!^{2}}{n!^{2}} . \frac{(N + n + 1)}{(1 − |a|)^{2N+2n+2}} . t^{2n−2})^{1/2} = ∥β∥_{2} . tc_{3}(N, a, t) ,

where c_{3}(N, a, t) tends to a finite limit as t → 0. Consequently,

σ_{2} ≤ c_{4}(M, N, a, t) . ∥α∥_{2} . ∥β∥_{2} . t ≤
≤ c_{4}(M, N, a, t) . ∥F ∥_{2} . ∥G∥_{2} . t ,

with c_{4}(M, N, a, t) tending to a finite limit as t → 0.

Similar estimate, of course, can be obtained for σ_{3}. Summing up,we see that

|⟨(R_{(M,N,a,t)} − T_{(M+1,N,a)} )F, G⟩| ≤ c_{5}(M, N, a, t) . t . ∥F ∥_{2} . ∥G∥_{2}

for all F, G ∈ A^{2}(D), where c_{5}(M, N, a, t) tends to a finite limit as t → 0. Consequently,

∥R_{(M,N,a,t)} − T_{(M+1,N,a)} ∥ ≤ c_{5}(M, N, a, t) . t

and the first part of the lemma follows. The assertion concerning R′_{(M,N,a,t)} can be proved in the same way. □

We need one more lemma, the proof of which is (fortunately) a little shorter. Remember the symbol ”clos” denotes the closure in the norm topology of B(A^{2}(D)).

Lemma 2.7. Denote T_{1} = {T_{ϕ} : ϕ ∈ D(D)}. Then

T_{(0,0,a)} ∈ closT_{1} for every a ∈ D .

Proof. For each δ in the interval (0, \frac{1−|a|}{2}), pick a function f_{δ} ∈ D(D) such that

f_{δ}(z) = \{ \begin{matrix}0 & if |z − a| ≥ δ + δ^{2} \\ δ^{−2} & if |z − a| ≤ δ \end{matrix}

and

0 ≤ f_{δ}(z) ≤ δ^{−2} if δ < |z − a| < δ + δ^{2} .

Let f, g ∈ a^{2}. Then

⟨(T_{f_{δ}} − T_{(0,0,a)} )f, g⟩ = ∫_{D} f_{δ}(z)f(z)\overline{g(z)}dz − f(a)\overline{g(a)} =

= [\frac{1}{δ^{2}} ∫_{|z−a|≤δ} f(z)\overline{g(z)}dz − f(a)\overline{g(a)}] + ∫_{δ<|z−a|<δ+δ^{2}} f_{δ}(z)f(z)\overline{g(z)}dz :=

:= ρ_{1} + ρ_{2} . (14)

Let f_{n}, resp. g_{n} be the coefficients of the Taylor expansion of f, resp. g at a:

f(x) = ∑_{0}^{∞}f_{n} . (x − a)^{n} , g(x) = ∑_{0}^{∞}g_{n} . (x − a)^{n} .

Substituting these formulas into the expression for ρ_{1} gives

ρ_{1} = \frac{1}{δ^{2}} ∫_{|z−a|≤δ} ∑_{m,n=0}^{∞} f_{n} . (z − a)^{n} . \overline{g}_{m} . \overline{(z − a)^{m}} dz − f_{0} \overline{g}_{0} =

= \frac{1}{δ^{2}} ∫_{0}^{δ} ∫_{0}^{2π} ∑_{m,n=0}^{∞} f_{n}\overline{g}_{m}r^{n+m} e^{(n−m)it} . \frac{r}{π} dt dr − f_{0} \overline{g}_{0} =

= \frac{1}{δ^{2}} . ∑_{n=0}^{∞}f_{n}\overline{g}_{n} . \frac{δ^{2n+2}}{n + 1} − f_{0} \overline{g}_{0} =

= ∑_{n=1}^{∞}f_{n}\overline{g}_{n} . \frac{δ^{2n}}{n + 1} .

Denote, for a little while,

α_{n} = \frac{(1 − |a|)^{n+1}}{(n + 1)^{1/2}} |f_{n}| , β_{n} = \frac{(1 − |a|)^{n+1}}{(n + 1)^{1/2}} |g_{n}| .

In course of the proof of Lemma 2.6, we have seen that α and β belong to l^{2} and

∥α∥_{2} ≤ ∥f∥_{2} , ∥β∥_{2} ≤ ∥g∥_{2} .

Now

∑_{n=1}^{∞}|f_{n}\overline{g}_{n}| . \frac{δ^{2n}}{n + 1} = ∑_{n=1}^{∞}α_{n}β_{n} . \frac{δ^{2n}}{(1 − |a|)^{2n+2}} =

= \frac{δ^{2}}{(1 − |a|)^{4}} . ∑_{n=1}^{∞}α_{n}β_{n} . [\frac{δ}{1 − |a|}]^{2(n−1)} ≤ \frac{δ^{2}}{(1 − |a|)^{4}} . ∑_{n=1}^{∞}α_{n}β_{n} ≤

≤ \frac{δ^{2}}{(1 − |a|)^{4}} . ∥α∥_{2} . ∥β∥_{2} ≤ \frac{δ^{2}}{(1 − |a|)^{4}} . ∥f∥_{2} . ∥g∥_{2} ,

which implies that

(15)  |ρ_{1}| ≤ \frac{δ^{2}}{(1 − |a|)^{4}} . ∥f∥_{2} . ∥g∥_{2} .

As for ρ_{2}, we have

|ρ_{2}| ≤ sup_{δ<|z−a|<δ+δ^{2}} |f_{δ}(z)f(z)\overline{g(z)}| .  ∫_{δ<|z−a|<δ+δ^{2}} dz ≤

≤ \frac{1}{δ^{2}} . sup_{δ<|z−a|<δ+δ^{2}} |f(z)g(z)| . δ^{3}(δ + 2) .

Because

|f(z)| = |⟨f, g_{z}⟩| ≤ ∥f∥_{2}. ∥g_{z}∥_{2} = ∥f∥_{2}. [g_{z}(z)]^{1/2} = \frac{∥f∥_{2}}{1 − |z|^{2}}

(and similarly for g), the supremum does not exceed

\frac{∥f∥_{2}. ∥g∥_{2}}{[1 − (|a| + δ + δ^{2})^{2}]^{2}} .

Summing up, we have

(16)  |ρ_{2}| ≤ δ . ∥f∥_{2}. ∥g∥_{2}. c_{6}(a, δ),

where

c_{6}(a, δ) = \frac{δ + 2}{[1 − (|a| + δ + δ^{2})^{2}]^{2}}

tends to a finite limit as δ → 0+.

Putting together (14),(15) and (16) yields

∥ T_{f_{δ}} − T_{(0,0,a)} ∥ ≤ c_{7}(a, δ) . δ ,

where c_{7}(a, δ) tends to a finite limit as δ → 0+. The lemma follows immediately. □

Proof. Second (constructive) proof (of Theorem 2.4). Note that the mapping ϕ ↦ T_{ϕ} is linear, so T_{1} and closT_{1} are linear subsets (i.e. subspaces) of B(A^{2}(D)). In view of Lemma 2.7, T_{(0,0,a)} ∈ closT_{1} for each a ∈ D. By linearity, R_{(0,0,a,t)} and R′_{(0,0,a,t)} ∈ closT_{1} whenever a ∈ D and |t| < 1 − |a|; by Lemma 2.6, this implies T_{(1,0,a)} and T_{(0,1,a)} ∈ closT_{1}. Proceeding by induction, we conclude that T_{(m,n,a)} ∈ closT_{1} for every a ∈ D and m, n = 0, 1, 2, . . .. Taking a = 0 shows that, in particular, ⟨. , z^{m}⟩z^{n} ∈ closT_{1}. By linearity, ⟨. , p⟩q ∈ closT_{1} whenever p, q are polynomials. Because polynomials are dense in A^{2}(D), necessarily ⟨. , f⟩g ∈ closT_{1} for all f, g ∈ A^{2}(D), i.e. all one-dimensional operators are in closT_{1}. Using the linearity of closT_{1} for the third time shows that all finite rank operators belong to closT_{1}; since these are dense in Comp (A^{2}(D)), Theorem 2.4 follows. □

Remark 2.8. For the Fock space A^{2}(C^{N}) in place of A^{2}(D), the proof comes through essentially without alterations. The Taylor series (6) are convergent for all x ∈ C, and so the formulas obtained from them are valid for all t ∈ C (instead of |t| < 1 − |a|). For the same reason, (9) becomes

∥F ∥_{2}^{2} = ∑_{n=0}^{∞}n!2^{n}|F_{n}|^{2},

and so ∥F ∥_{2} = ∥α∥_{2} this time, where α ∈ l^{2},

α_{k} = (k!2^{k})^{1/2}|F_{k}|,

and similarly for G. Formula (11) becomes

∥α∥_{2}. ( ∑_{m=1}^{∞} \frac{(M + m)!^{2}}{m!^{2}} . \frac{t^{2m−1}}{(M + m)!.2^{M+m}} )^{1/2} =

= ∥α∥_{2}. (t ∑_{m=0}^{∞} \frac{(M + m + 1)!}{(m + 1)!^{2} . 2^{M+m+1}} t^{2m})^{1/2} =

= ∥F ∥_{2} . t^{1/2}c_{1}(M, t),

where c_{1}(M, t) tends to a finite limit as t^{2} → 0+, i.e. as t → 0; the estimate (12) for σ_{1}

σ_{1} ≤ ∥F ∥_{2}.∥G∥_{2}.tc_{2}(M, N, t)

follows. Similarly, (13) becomes

∥β∥_{2}. (∑_{n=2}^{∞} \frac{(N + n)!}{n!^{2}} \frac{t^{2n−2}}{2^{N+n}} )^{1/2} = ∥G∥_{2}.tc_{3}(N, t),

with

c_{3}(N, t) = (∑_{n=0}^{∞} \frac{N + n + 2)!}{(n + 2)!^{2}} \frac{t^{2n}}{2^{N+n}} )^{1/2}

tending to a finite limit as t → 0, and so

σ_{2} ≤ c_{4}(M, N, t).∥F ∥_{2}.∥G∥_{2}.t

with c_{4}(M, N, t) tending to a finite limit as t → 0.

In the proof of Lemma 2.7, the functions f_{δ} ∈ D(C) must be chosen so that

f_{δ}(z) = 0 if |z − a| ≥ δ + δ^{2},
f_{δ}(z) = 2δ^{−2} if |z − a| ≤ δ,

and 0 ≤ f_{δ}(z) ≤ 2/δ^{2} otherwise; here 0 < δ < 1. Proceed till you define ρ_{1} and ρ_{2} as in (14); then pass to polar coordinates to obtain

ρ_{1} = \frac{2}{δ^{2}} ∫_{0}^{δ} \frac{1}{2π} ∫_{0}^{2π} ∑_{n=0}^{∞} ∑_{m=0}^{∞} f_{n}\overline{g_{m}}r^{n+m} e^{(n−m)it}e^{−r^{2} /2} dt r dr − f_{0} \overline{g_{0}} =

= \frac{1}{δ^{2}} ∑_{n=0}^{∞}2^{n+1} f_{n}\overline{g_{n}} ∫_{0}^{δ^{2} /2} t^{n}e^{−t} dt − f_{0} \overline{g_{0}}.

For n ≥ 1, the function t^{n}e^{−t} increases on the interval (0, n) ⊃ (0, δ); consequently, ∫_{0}^{δ^{2} /2} t^{n}e^{−t} dt ≤ (δ^{2}/2)^{n+1} e^{−δ^{2} /2}, and

|ρ_{1}| ≤ |\frac{2}{δ^{2}} f_{0} \overline{g_{0}}.(1 − e^{−δ^{2} /2}) − f_{0} \overline{g_{0}} | + |\frac{2}{δ^{2}} e^{−δ^{2} /2} ∑_{n=1}^{∞}2^{n+1} f_{n}\overline{g_{n}} (\frac{δ^{2}}{2} )^{n+1} | =

= |f_{0} \overline{g_{0}}|.O (\frac{δ^{2}}{2} ) + 2e^{−δ^{2} /2}. ∑_{n=1}^{∞}|f_{n}\overline{g_{n}}|δ^{2n}.

Employing the α, β ∈ l^{2} again, the last expression is seen to be bounded by

c_{6}(δ).δ^{2}.∥F ∥_{2}.∥G∥_{2},

with c_{6}(δ) tending to a finite limit as δ → 0+.

As for ρ_{2}, we have, again,

|ρ_{2}| ≤ sup_{δ<|z−a|<δ+δ^{2}} |f_{δ}(z)f(z)\overline{g(z)}| .  ∫_{δ<|z−a|<δ+δ^{2}} dz ≤

≤\frac{2}{δ^{2}} sup_{δ<|z−a|<δ+δ^{2}} |f(z)g(z)| . \frac{δ^{3}(δ + 2)}{2} e^{−δ^{2} /2} ≤
≤δ(δ + 2)e^{−δ^{2} /2} sup_{δ<|z−a|<δ+δ^{2}} |f(z)g(z)|,

sup_{δ<|z−a|<δ+δ^{2}} |f(z)g(z)| ≤ ∥F ∥_{2}.∥G∥_{2}.e^{(δ+δ^{2} )^{2} /2} ,

|ρ_{2}| ≤ δ(δ + 2).∥F ∥_{2}.∥G∥_{2}.e^{δ^{3} (δ+2)/2} = δ.∥F ∥_{2}.∥G∥_{2}.c_{6}(δ),

where c_{6}(δ) → 3 as δ → 0+. Thus,

|ρ_{1} + ρ_{2}| ≤ ∥F ∥_{2}.∥G∥_{2}.δ.c_{7}(δ),

where c_{7}(δ) → 3 as δ → 0+, and Lemma 2.7 follows as before. This shows that the second proof of Theorem 2.4 works also for the Fock space A^{2}(C^{N}). □

Remark 2.9. Because Comp (A^{2}(Ω) ) is SOT-dense in B(A^{2}(Ω) ) and norm convergence implies SOTconvergence, Theorem 2.4 yields another proof of Theorem 2.3. □

For Ω = D, there is a natural intermediate function space between D(D) and L^{∞}(D) — namely, C(\overline{D}), the functions continuous on the closed unit disc \overline{D}. The norm closure of the set T_{2} = \{T_{ϕ} : ϕ ∈ C(\overline{D})\} was described in my paper [16], using the techniques of Olin and Thomson and of Bunce.

Theorem 2.10. closT_{2} is a C^{∗}-algebra and coincides with T_{2} + Comp (A^{2}(D)).

Corollary 2.11. For every T ∈ closT_{2}, σ_{e}(T) is connected. In particular, closT_{2} ≠ B(A^{2}(D)).

We omit the proofs, which can be found in [16]. Using our Theorem 2.4, they can also be deduced from results of Axler, Conway and McDonald [3].

Chapter 3. TOEPLITZ ALGEBRAS

As we have already mentioned, the results of the previous chapter prompt to ask whether the Toeplitz operators (e.g. on A^{2}(D)) are not actually norm-dense in the space of all linear operators; or if, at least, the C^{∗}-algebra generated by them is not all of B(A^{2}(D)). In this chapter, we are going to prove that this is not the case: all Toeplitz operators on the Bergman space A^{2}(D) are contained in a certain C^{∗}-algebra which is a proper subalgebra of B(A^{2}(D)). The same result holds also for the classical Hardy space H^{2}, the Fock space A^{2}(C), Bergman spaces A^{2}(Ω) for a wide class of domains Ω ⊂ C and for spaces H^{2}(ρ) (to be defined later in this chapter). Moreover, the corresponding algebras can be chosen to be, loosely speaking, the same (more precisely: spatially isomorphic to each other) in all these cases. Let us start with A^{2}(D), where the proof is most transparent.

Define

A(T_{z}) := {T ∈ B(A^{2}(D)) : T − T_{z}^{∗}T T_{z} ∈ Comp }.

There is an alternative definition of A(T_{z}):

Proposition 3.1. A(T_{z}) = {T ∈ B(A^{2}(D)) : [T, T_{z}] ∈ Comp }.

Proof. The operators

I − T_{z}^{∗}T_{z} = diag(1 − \frac{n + 1}{n + 2} ) = diag(\frac{1}{n + 2} )

and

I − T_{z}T_{z}^{∗} = diag(1 − \frac{n}{n + 1} ) = diag(\frac{1}{n + 1} )

are compact; consequently,

T − T_{z}^{∗}T T_{z} ∈ Comp⟹ T_{z}(T − T_{z}^{∗}T T_{z}) = (T_{z}T − T T_{z}) + (I − T_{z}T_{z}^{∗})T T_{z} ∈ Comp⟺
⟺ T_{z}T − T T_{z} ∈ Comp ,

and, on the other hand,

T_{z}T − T T_{z} ∈ Comp⟹ T_{z}^{∗}T_{z}T − T_{z}^{∗}T T_{z} = (T − T_{z}^{∗}T T_{z}) − (I − T_{z}^{∗}T_{z})T ∈ Comp⟺
⟺ T − T_{z}^{∗}T T_{z} ∈ Comp .

□ The following theorem shows that A(T_{z}) is the algebra appealed to above.

Theorem 3.2. (i) A(T_{z}) is a C^{∗}-algebra.

(ii) ∀ϕ ∈ L^{∞}(D) : T_{ϕ} ∈ A(T_{z}).

Proof. (i) It’s clear that A(T_{z}) is a linear and selfadjoint set, which is moreover closed in the norm topology; so the only thing that remains to be checked is that it is closed under multiplication. But

[AB, T_{z}] = A(BT_{z} − T_{z}B) + (AT_{z} − T_{z}A)B = A[B, T_{z}] + [A, T_{z}]B,

which is compact if [A, T_{z}] and [B, T_{z}] are.

(ii) If ϕ ∈ L^{∞}(D), then

T_{ϕ} − T_{z}^{∗}T_{ϕ}T_{z} = T_{ϕ} − T\overline{z}T_{ϕ}T_{z} = T_{ϕ−\overline{z}ϕz} = T_{(1−|z|^{2} )ϕ(z)} .

But (1 − |z|^{2})ϕ(z) ∈ V (D) and so the last operator is compact by Corollary 1.5. □

Corollary 3.3. The C^{∗}-algebra generated by {T_{ϕ} : ϕ ∈ L^{∞}(D)} is strictly smaller than B(A^{2}(D)).

Proof. In view of the preceding Theorem, it suffices to find an operator not in A(T_{z}); one of them is

J = diag(−1)^{n},

since

J − T_{z}^{∗}JT_{z} = diag((−1)^{n} − \frac{n + 1}{n + 2} (−1)^{n+1} )

certainly is not compact. □

Theorem 3.2 carries over trivially to the classical Hardy space H^{2}. Indeed, when T_{ϕ} is a Toeplitz operator on H^{2}, then

T_{ϕ} = S^{∗}T_{ϕ}S,

where S = T_{z} is the usual (forward) shift operator on H^{2}. Thus, if we define

A(S) := {T ∈ B(H^{2}) : T − S^{∗}T S ∈ Comp (H^{2})},

then the following assertions are immediate.

Proposition 3.4. A(S) = {T ∈ B(H^{2}) : [T, S] ∈ Comp (H^{2})}.

Theorem 3.5. (i) A(S) is a C^{∗}-subalgebra of B(H^{2}). (ii) T_{ϕ} ∈ A(S) for every Toeplitz operator T_{ϕ} on H^{2}.

Corollary 3.6. The C^{∗}-algebra generated by the Toeplitz operators in B(H^{2}) is strictly smaller than B(H^{2}).

The proofs are similar to those for 3.1 — 3.3, and actually a lot simpler. In the Corollary, the same operator J (this time, of course, diagonality is understood with respect to the standard orthonormal basis {z^{n}}_{n=0}^{∞} of H^{2}) works. The algebras A(T_{z}) and A(S) are, in fact, isomorphic; moreover, the isomorphism A(T_{z}) → A(S) may be chosen to be spatial, i.e. of the form

T ↦ W^{∗}T W,

where W is a fixed unitary operator from H^{2} onto A^{2}(D). To see this, let W be the operator mapping the standard basis \{z^{n}\}_{n∈N} of H^{2} onto the basis \{\sqrt{n + 1} z^{n}\}_{n∈N} od A^{2}(D),

W : ∑_{n=0}^{∞}f_{n}z^{n} ↦ ∑_{n=0}^{∞}f_{n}\sqrt{n + 1} z^{n}.

Then

T ∈ A(T_{z}) ⟺ [T, T_{z}] ∈ Comp⟺ W^{∗}T T_{z}W − W^{∗}T_{z}T W ∈ Comp⟺
⟺ (W^{∗}T W)(W^{∗}T_{z}W) − (W^{∗}T_{z}W)(W^{∗}T W) ∈ Comp⟺
⟺ (W^{∗}T W)S − S(W^{∗}T W) ∈ Comp⟺ W^{∗}T W ∈ A(S);

here T_{z} is the Toeplitz operator on A^{2}(D), not on H^{2}, and the last-but-one equivalence is due to the fact that

W^{∗}T_{z}W − S = S.diag(\sqrt{\frac{n + 1}{n + 2}} − 1)

is a compact operator^{4}.

In general, we may define

A(M) := {T ∈ B(H) : [M, T] ∈ Comp (H)}

4Here, once more, the diagonality is understood with respect to the standard basis of H^{2}. for arbitrary operator M on a Hilbert space H. The following theorem generalizes the considerations of the previous paragraph.

Theorem 3.7. (i) A(M) = A(M + K) for arbitrary compact operator K.

(ii) Suppose that σ_{e}(M) = T and indM = −1 (the Fredholm index). Then there exists a unitary operator W : H^{2} → H such that the transformation

T ↦ W^{∗}T W

is a C^{∗}-algebra isomorphism of A(M) onto A(S). In particular, A(M) is a proper C^{∗}-subalgebra of B(H).

Proof. (i) is immediate (actually, it has already been used in the end of the last-but-one paragraph). (ii) According to the Brown-Douglas-Fillmore theory [8], an operator M satisfying these conditions is

unitarily equivalent to S modulo the compacts, i.e. there exists a unitary operator W : H^{2} → H and a compact operator K ∈ Comp (H) such that

W SW^{∗} = M + K.

Owing to part (i), A(M) = A(W SW^{∗}), and rehearsing the argumentation of the last-but-one paragraph leads to the desired conclusion. □

Now we are in a position to prove the analogue of Theorem 3.2 for general Bergman spaces A^{2}(Ω) . In case the domain Ω is bounded, a short proof may be given based on the results of Axler, Conway & McDonald [3] or of Olin & Thomson [23]^{5}. We shall present it first, and then, in case the domain Ω is simply connected, employ an idea of Axler to obtain a more elementary proof which uses only methods of complex function theory.

We begin by recalling the pertinent results of Axler et al. [3]. Assume that Ω is bounded. A point x ∈ ∂Ω is called removable if there exists a neighbourhood V of x such that every function f ∈ A^{2}(Ω) can be analytically continued to V. (For instance, every isolated point of ∂Ω is removable, by a variant of Riemann’s removable singularity theorem.) The collection of all removable boundary points is called ∂_{r}Ω, the removable boundary of Ω; ∂_{e}Ω := ∂Ω \ ∂_{r}Ω is the essential boundary . It is proved in [3] that Ω ∪ ∂_{r}Ω is an open set and that ∂_{r}Ω has zero Lebesgue measure; consequently, L^{2}(Ω) = L^{2}(Ω ∪ ∂_{r}Ω) and A^{2}(Ω) = A^{2}(Ω ∪ ∂_{r}Ω). This makes it possible to replace, without loss of generality, Ω by Ω ∪ ∂_{r}Ω, i.e. to assume that ∂_{r}Ω = ∅, ∂Ω = ∂_{e}Ω. In that case, the following two assertions hold.

Proposition 3.8. (cf. [3], Prop. 8) If^{6} f ∈ C(\overline{Ω} ), then H_{f} ∈ Comp (A^{2}(Ω) ).

Proposition 3.9. ([3], Corol. 10) If f ∈ C(\overline{Ω} ), then σ_{e}(T_{f}) = f(∂Ω).

Now we are ready to prove our main theorem.

Theorem 3.10. Assume that Ω is a bounded domain in C. Then there exists a unitary operator W : H^{2} → A^{2}(Ω) such that the transformation

T ↦ W^{∗}T W, B(A^{2}(Ω) ) → B(H^{2}),

sends every Toeplitz operator T_{ϕ}, ϕ ∈ L^{∞}(Ω) , on A^{2}(Ω) to an element of A(S). In particular, the C^{∗}-algebra generated by the Toeplitz operators on A^{2}(Ω) is a proper subalgebra of B(A^{2}(Ω) ).

Proof. Without loss of generality we may assume D ⊂ Ω. Let Φ ∈ L^{∞}(Ω) be the function z/|z| adjusted in a small neighbourhood of 0 so as to be continuous on \overline{Ω} ; for instance, take

Φ(re^{it}) = \{ \begin{matrix}e^{it} & if r ≥ 1, \\ re^{it} & if r ≤ 1. \end{matrix}

5The main idea, however, goes back to Bunce[9]. 6C(\overline{Ω} ) is the space of functions continuous on the closure \overline{Ω} of Ω.

Because Φ ∈ C(\overline{Ω} ), the Hankel operators H_{Φ} and H\overline{Φ} are compact by Proposition 3.8 and, consequently, so must be the operator (cf. Proposition 1.2)

[T_{ϕ}, T_{Φ} ] = H∗ ; \overline{Φ}H_{ϕ} − H∗ ; \overline{ϕ}H_{Φ}

for arbitrary ϕ ∈ L^{∞}(Ω) ; thus, T_{ϕ} ∈ A(T_{Φ} ). In view of Proposition 3.9, σ_{e}(T_{Φ} ) = Φ(∂Ω) = T. If we prove that indT_{Φ} = −1, we can apply Theorem 3.7 and the desired conclusion will follow.

For 0 ≤ θ ≤ 1, define

Φ_{θ}(re^{it}) = \{ \begin{matrix}r^{θ}e^{it} & for r ≥ 1, \\ re^{it} & for r ≤ 1. \end{matrix}

Then Φ_{θ} ∈ L^{∞}(Ω) , and so the operators T_{Φ_{θ}} are defined. Moreover, for arbitrary θ_{1}, θ_{2} ∈ ⟨0, 1⟩,

∥T_{Φ_{θ_{1}}} − T_{Φ_{θ_{2}}} ∥ ≤ ∥Φ_{θ_{1}} − Φ_{θ_{2}} ∥_{∞} = sup_{1≤r≤diam Ω} |r^{θ_{1}} − r^{θ_{2}}| ≤ c.|θ_{1} − θ_{2}|,

where diamΩ is the diameter of Ω and

c = sup_{\begin{matrix}1≤r≤diam Ω \\ 0≤θ≤1 \end{matrix}} |r^{θ} lnr|

does not depend on θ. This shows that the mapping θ ↦ T_{Φ_{θ}} is continuous. Besides, σ_{e}(T_{Φ_{θ}} ) = Φ_{θ}(∂_{e}Ω) ∌ 0, i.e. all T_{Φ_{θ}} are Fredholm and so their index is defined. Since ”ind” is a continuous integer-valued function, it must be constant along the path θ ↦ T_{Φ_{θ}} , whence indT_{Φ_{θ_{0}}} = indT_{Φ_{θ_{1}}} , or

indT_{Φ} = indT_{z}.

But kerT_{z} = {0}, while RanT_{z} consists of all functions from A^{2}(Ω) that vanish at 0. Consequently, indT_{z} = −1, and the proof is complete. □

If Ω ⊂ C is simply connected, it turns out that the condition that Ω be bounded may be weakened — namely, it suffices that Ω have finite Lebesgue measure. (If the latter condition is not met, the spaces A^{2}(Ω) become too small — they won’t even contain nonzero constant functions.) It is also not necessary to appeal to the results quoted above if one is willing to do a little computing. So suppose Ω⫋C is simply connected and let Φ : Ω → D be the Riemann mapping function.

Lemma 3.11. σ_{e}(T_{Φ} ) = T and indT_{Φ} = −1.

Proof. For arbitrary x ∈ D, the operator T_{Φ−x} is injective and its range clearly consists exactly of functions vanishing at Φ^{−1}(x) (since Φ(z) − x, loosely speaking, behaves like z − Φ^{−1}(x) in a sufficiently small neighbourhood of Φ^{−1}(x)). Hence, indT_{Φ−x} = −1 for x ∈ D. On the other hand, ∥T_{Φ} ∥ ≤ ∥Φ∥_{∞} = 1 and so indT_{Φ−x} = 0 if |x| > 1. Since

x ↦ indT_{Φ−x}

is a continuous function on C \ σ_{e}(T_{Φ} ), necessarily σ_{e}(T_{Φ} ) = T. □

Lemma 3.12. Assume T is a Fredholm operator, indT < 0, and I − T^{∗}T ∈ Comp . Then also I − T T^{∗} ∈ Compand T is essentially normal.

Proof. (Cf. [8], proof of Theorem 3.1.) By assumption, I − T^{∗}T ∈ Comp , and on multiplying by the inverse of I + (T^{∗}T)^{1/2}, we find that I − (T^{∗}T)^{1/2} ∈ Comp . If T = W (T^{∗}T)^{1/2} is the polar decomposition, it follows that T is a compact perturbation of the partial isometry W. Since indW = indT < 0, dimkerW < codimRanW, there exists a partial isometry L with initial space kerW and final space contained in (RanW)^{⊥}, and therefore T is a compact perturbation of the isometry V = W + L. Now I − V V^{∗} is the projection onto (RanV)^{⊥}, which is a subspace of dimension −indV = −indT < +∞; hence, I − V V^{∗} is a finite rank operator. Because T is a compact perturbation of V, I − T T^{∗} must be a compact operator. □

Theorem 3.14. Assume Ω ⊂ C is a simply connected domain (of finite Lebesgue measure) and let Φ : Ω → D be the Riemann mapping function. Then T_{f} ∈ A(T_{Φ} ) ∀f ∈ L^{∞}(Ω) and there exists a unitary operator W : H^{2} → A^{2}(Ω) such that the transformation T ↦ W^{∗}T W establishes a C^{∗}-isomorphism of A(T_{Φ} ) onto A(S). In particular, the C^{∗}-algebra generated by the Toeplitz operators T_{f}, f ∈ L^{∞}(Ω) , is a proper subalgebra of B(A^{2}(Ω) ).

Proof. Let f ∈ L^{∞}(Ω) . We have
T_{f} − T_{Φ}^{∗}T_{f}T_{Φ} = T_{f−\overline{Φ} fΦ} = T_{(1−|Φ|^{2} )f} .

For arbitrary δ > 0, the set

{y ∈ Ω : |Φ(y)| ≤ 1 − δ}

is a compact subset of Ω. Consequently, (1 − |Φ|^{2})f ∈ V (Ω), and Proposition 1.4 implies that

(17)  T_{f} − T_{Φ}^{∗}T_{f}T_{Φ} ∈ Comp ∀ f ∈ L^{∞}(Ω) .

Taking f = 1, we see that I − T_{Φ}^{∗}T_{Φ} must be compact, and an application of Lemma 3.11 and Lemma 3.12 shows that I − T_{Φ} T_{Φ}^{∗} is compact as well. Multiplying (17) by T_{Φ} from the left yields (cf. the proof of Proposition 3.1)

T_{Φ} T_{f} − T_{f}T_{Φ} ∈ Comp ,

i.e. T_{f} ∈ A(T_{Φ} ). It remains only to make use of Lemma 3.11, Lemma 3.12 and Theorem 3.7. □

The Theorems 3.2, 3.5, 3.10 & 3.14 imply that the same situation is encountered for A^{2}(D), H^{2} and A^{2}(Ω) for Ω bounded (not necessarily simply connected) or for Ω simply connected and of finite measure. Prior to investigating what happens for the Fock space, we briefly discuss another class of spaces which also seem to join here.

Let ρ be a positive Borel measure on ⟨0, 1⟩. Set

a_{n} = ∫_{⟨0,1⟩} t^{n} dρ(t)

and define H^{2}(ρ) to be the space of all functions f(z) = ∑_{n=0}^{∞} f_{n}z^{n} analytic on D for which

∥f∥_{ρ} := (∑_{n=0}^{∞}a_{n}|f_{n}|^{2})^{1/2} < +∞.

Example: When dρ(r) = dr (the Lebesgue measure on ⟨0, 1⟩), H^{2}(ρ) is our old friend A^{2}(D). The following propositions summarize basic properties of these spaces.

Proposition 3.18. (i) H^{2}(ρ) is a Hilbert space with scalar product

⟨f, g⟩_{ρ} := ∑_{n=0}^{∞}a_{n}f_{n}\overline{g_{n}}, where f(z) = ∑_{n=0}^{∞}f_{n}z^{n}, g(z) = ∑_{n=0}^{∞}g_{n}z^{n}.

The set \{z^{n}/\sqrt{a_{n}}\}_{n∈N} is an orthonormal basis for H^{2}(ρ).

(ii) If ρ({1}) = 0 (i.e. ρ has no mass at 1), then

∥f∥_{ρ}^{2} = ∫_{⟨0,1⟩} ∫_{0}^{2π} |f(r^{1/2}e^{it})|^{2} \frac{dt}{2π} dρ(r)

for any function f analytic on D. If ρ({1}) > 0, the last formula still holds if we agree to set

∫_{0}^{2π} |f(e^{it})|^{2} \frac{dt}{2π} = sup_{0<r<1} ∫_{0}^{2π} |f(re^{it})|^{2} \frac{dt}{2π}.

Proof. (i) is immediate; in fact, the mapping defined by f ↦ \{\sqrt{a_{n}}f_{n}\}_{n∈N} is an isometric isomorphism of H^{2}(ρ) onto l^{2}(N). (ii) Pass to the polar coordinates and compute to get

∫_{⟨0,R)} ∫_{0}^{2π} |f(r^{1/2}e^{it})|^{2} \frac{dt}{2π} dρ(r) = ∑_{n=0}^{∞}(∫_{⟨0,R)} r^{n} dρ(r)).|f_{n}|^{2}.

Letting R ↗ 1 gives the desired formula. (Interchanges of integration and summation signs can be justified

by the uniform convergence of ∑_{n=0}^{∞} f_{n}z^{n} on compact subsets of D, and by the Lebesgue monotone convergence

theorem.) □

Proposition 3.19. The evaluation functionals f ↦ f(λ), λ ∈ D, on H^{2}(ρ) are continuous for all λ ∈ D if and only if 1 ∈ suppρ.

Proof. Since f(λ) = ∑_{n=0}^{∞} f_{n}λ^{n} = ∑_{n=0}^{∞} a_{n}f_{n}\overline{(\frac{\overline{λ}^{n}}{a_{n}} )}, the evaluation functional at λ ∈ D is continuous on

H^{2}(ρ) iff the function

g_{λ}(z) = ∑_{n=0}^{∞} \frac{\overline{λ}^{n}z^{n}}{a_{n}}

belongs to H^{2}(ρ). Because ∥g_{λ}∥_{ρ}^{2} = ∑_{n=0}^{∞} \frac{|λ|^{2n}}{a_{n}} , this happens iff the radius of convergence of the series ∑_{n=0}^{∞} \frac{x^{n}}{a_{n}}

is at least one, i.e. iff

lim sup|a_{n}|^{1/n} ≥ 1.

The expression a_{n}^{1/n} = (∫_{⟨0,1⟩} t^{n} dρ(t))^{1/n} is the norm of the function t ↦ t in the Lebesgue space L^{n}(⟨0, 1⟩, dρ). For ρ finite (which is our case) this is well-known to tend to the norm in L^{∞}(⟨0, 1⟩, dρ), i.e. to ess supt = sup{t : t ∈ suppρ}. This is ≥ 1 if and only if 1 ∈ suppρ. □

In order to define Toeplitz operators on H^{2}(ρ), we need it to be a subspace of ”some L^{2}”. At that moment, technical difficulties arise when ρ has positive mass at 1, i.e. when ρ({1}) > 0; so let us agree to exclude this case. Also, replacing ρ by cρ for some number c > 0 leads to isomorphic spaces H^{2}(ρ) and H^{2}(cρ); so there is no harm in assuming ρ(⟨0, 1⟩) = 1. Thus we are lead to the condition

(20)  ρ({1}) = 0,  ρ(⟨0, 1⟩) = 1,  1 ∈ suppρ,

whose validity shall be assumed from now on. Proposition 3.18(ii) then implies that H^{2}(ρ) is a closed subspace of the Hilbert space

L^{2}(ρ) := \{f on D : ∫_{⟨0,1⟩} ∫_{0}^{2π} |f(r^{1/2}e^{it})|^{2} \frac{dt}{2π} dρ(r)\} < +∞

(with the obvious inner product). Let P_{+} be the orthogonal projection of L^{2}(ρ) onto H^{2}(ρ) and define Toeplitz operator T_{ϕ} : H^{2}(ρ) → H^{2}(ρ) by

T_{ϕ}f := P_{+} ϕf, f ∈ H^{2}(ρ),

where ϕ ∈ L^{∞}(ρ) := L^{∞}(D, dt dρ(r)). The formulas from Proposition 1.2

T_{ϕ}T_{g} = T_{ϕg}, T\overline{g}T_{ϕ} = T_{\overline{g}ϕ}, for ϕ ∈ L^{∞}(ρ), g ∈ H^{∞}(D) ⊂ L^{∞}(ρ),

are readily seen to remain in force, as well as Corollary 1.5. (The proofs carry over without change.) Consequently,

(21)  T_{ϕ} − T_{z}^{∗}T_{ϕ}T_{z} = T_{(1−|z|^{2} )ϕ} ∈ Comp (H^{2}(ρ)).

The following proposition implies that this is equivalent to T_{ϕ} ∈ A(T_{z}) and that A(T_{z}) is, once again, C^{∗}-isomorphic to A(S).

Proposition 3.20. Assume that the condition (20) is fulfilled. Let Z be the (forward) shift operator on H^{2}(ρ) with respect to the orthonormal basis \{z^{n}/\sqrt{a_{n}}\}_{n∈N} . Then T_{z} is a compact perturbation of Z.

Proof. Denote, for a little while, e_{n} = z^{n}/\sqrt{a_{n}}, so that Ze_{n} = e_{n+1} (n ∈ N). By definition,

T_{z}z^{n} = z^{n+1} , i.e. T_{z}e^{n} = \sqrt{\frac{a_{n+1}}{a_{n}}} e_{n+1} and Z − T_{z} = Z.diag(1 − \sqrt{\frac{a_{n+1}}{a_{n}}} ). Hence it suffices to show that

\frac{a_{n+1}}{a_{n}} → 1.

Clearly a_{n+1} ≤ a_{n} and, by Hölder’s inequality, a_{n}^{1/n} ≤ a_{m}^{1/m} if m > n. Taking m = n + 1 gives

an^{\frac{n+1}{n}} ≤ a_{n+1} ≤ a_{n},

or

a_{n}^{1/n} ≤ \frac{a_{n+1}}{a_{n}} ≤ 1.

In course of the proof of Proposition 3.19, the left-hand side was observed to tend to sup(suppμ) = 1 as n → ∞. The proposition follows. □

Corollary 3.21. Assume that (20) is fulfilled. Then T_{ϕ} ∈ A(T_{z}) ∀ϕ ∈ L^{∞}(ρ), and A(T_{z}) = A(Z) is spatially isomorphic to A(S).

Proof. Premultiplying (21) by T_{z} shows that

T_{z}T_{ϕ} − T_{z}T_{z}^{∗}T_{ϕ}T_{z} = [T_{z}, T_{ϕ}] + (I − T_{z}T_{z}^{∗})T_{ϕ}T_{z}

is compact; owing to the last proposition, I − T_{z}T_{z}^{∗} ∈ Comp , and so also [T_{z}, T_{ϕ}] ∈ Comp , i.e. T_{ϕ} ∈ A(T_{z}). By Theorem 3.7(i), A(T_{z}) = A(Z). Finally, if we define W : H^{2} → H^{2}(ρ) by z^{n} ∈ H^{2} ↦ z^{n}/\sqrt{a_{n}} ∈ H^{2}(ρ), then Z = W SW^{∗}, and so the transformation T ∈ A(Z) ↦ W^{∗}T W ∈ A(S) is a C^{∗}-algebra isomorphism of A(Z) onto A(S). □

Corollary 3.22. Assume that (20) is fulfilled. Then the C^{∗}-algebra generated by T_{ϕ}, ϕ ∈ L^{∞}(ρ), is a proper subalgebra of B(H^{2}(ρ)).

Remark 3.23. We conclude our brief excursion into H^{2}(ρ) spaces by a simple example. Take ρ to be the Lebesgue measure on ⟨0, 1⟩. Then

a_{n} = \frac{1}{n + 1} , n ∈ N,

H^{2}(ρ) is but our old friend A^{2}(D), and the last Corollary reduces to Theorem 3.2. □

Now let us turn our attention to the Fock space A^{2}(C). Recall that it has an orthonormal basis {e_{n}}_{n=0}^{∞} ,

e_{n}(z) := (n! 2^{n})^{−1/2} z^{n}.

Denote Z the forward shift operator with respect to this basis, and let

Φ(z) = \frac{z}{|z|} = e^{i arg z}.

Theorem 3.24. (i) The operator T_{Φ} is a compact perturbation of Z. Consequently, A(T_{Φ} ) = A(Z). (ii) T_{f} ∈ A(T_{Φ} ), i.e. T_{f}T_{Φ} − T_{Φ} T_{f} ∈ Comp , for every f ∈ L^{∞}(C). (iii) There exists a unitary operator W : H^{2} → A^{2}(C) such that the transformation T ↦ W^{∗}T W is a

C^{∗}-isomorphism of A(Z) onto A(S).

In particular, the C^{∗}-algebra generated by all T_{f}, f ∈ L^{∞}(C), is a proper subset of B(A^{2}(C)).

Proof. (i) Compute:

⟨T_{Φ} z^{n}, z^{m}⟩ = ∫_{C} \frac{z}{|z|} z^{n}\overline{z}^{m} dμ(z) = ∫_{0}^{+∞} ∫_{0}^{2π} r^{n+m} e^{(n−m+1)it} \frac{dt}{2π} e^{−r^{2} /2}r dr.

This is zero unless m = n + 1, and in that case it equals

∫_{0}^{+∞} r^{2n+1} e^{−r^{2} /2}r dr = ∫_{0}^{+∞} 2^{n+ \frac{1}{2}} t^{n+ \frac{1}{2}} e^{−t} dt = 2^{n+ \frac{1}{2}} Γ(n + \frac{3}{2}),

where Γ is Euler’s gamma-function. Thus

⟨T_{Φ} e_{n}, e_{m}⟩ = \{ \begin{matrix}0 & if m ≠ n + 1 \\ (n! 2^{n})^{−1/2}.(m! 2^{m})^{−1/2}.2^{n+ \frac{1}{2}} Γ(n + \frac{3}{2}) & if m = n + 1. \end{matrix}

Consequently, T_{Φ} e_{n} = c_{n}e_{n+1} , where

c_{n} = \frac{Γ(n + \frac{3}{2})}{Γ(n + 1)^{1/2}.Γ(n + 2)^{1/2}} .

It follows that

Z − T_{Φ} = Z . diag(1 − c_{n}),

and in order to verify our claim it suffices to show that c_{n} → 1 as n → +∞. According to Stirling’s formula,

Γ(x + 1) ∼ \sqrt{2π} x^{x+ \frac{1}{2}} e^{−x},

where ”∼” means that the ratio of the right-hand to the left-hand side approaches 1 as x → +∞. Substituting this into the expression for c_{n} produces

c_{n} ∼ \frac{(n + \frac{1}{2})^{n+1} . e^{−n− \frac{1}{2}} . \sqrt{2π}}{n^{\frac{n}{2}+ \frac{1}{4}} . e^{− \frac{n}{2}} . ^{4} \sqrt{2π} . (n + 1)^{\frac{n}{2}+ \frac{3}{4}} .e^{− \frac{n}{2}− \frac{1}{2}} . ^{4} \sqrt{2π}} .

The terms containing π cancel, as well as those containing e, and what remains is the product of

(\frac{n + \frac{1}{2}}{n} )^{n/2} ,  (\frac{n + \frac{1}{2}}{n + 1} )^{\frac{n+1}{2}}  and  \frac{(n + \frac{1}{2})^{1/2}}{n^{1/4}.(n + 1)^{1/4}} ,

which tend to e^{1/4}, e^{−1/4} and 1, respectively. So, indeed, c_{n} → 1 and the assertion follows.

(ii) Recall the formulas (cf. Proposition 1.6)

(22)  T_{fg} − T_{f}T_{g} = H∗ ; \overline{f}H_{g},  T_{f}T_{g} − T_{g}T_{f} = H∗ ; \overline{g}H_{f} − H∗ ; \overline{f}H_{g},

which hold for arbitrary f, g ∈ L^{∞}(C). Owing to the second one,

T_{f}T_{Φ} − T_{Φ} T_{f} = H∗ ; \overline{Φ}H_{f} − H∗ ; \overline{f}H_{Φ}

will be compact for arbitrary f ∈ L^{∞}(C) if H_{Φ} , H\overline{Φ} ∈ Comp . The latter is equivalent to H_{Φ}^{∗}H_{Φ} , H∗ ; \overline{Φ}H\overline{Φ} ∈ Comp , respectively, and the first formula in (22) shows that this in turn is equivalent to

I − T_{Φ}^{∗}T_{Φ} and I − T_{Φ} T_{Φ}^{∗} ∈ Comp ,

respectively. Owing to part (i), the last two operators are compact perturbations of I − Z^{∗}Z = 0 and I − ZZ^{∗} = ⟨ . , e_{0}⟩e_{0}, respectively, and the result follows.

(iii) Define W : H^{2} → A^{2}(C) by mapping the standard basis of H^{2} onto the basis {e_{n}}_{n∈N} of A^{2}(C),

W : z^{n} ∈ H^{2} ↦ \frac{z^{n}}{\sqrt{n! 2^{n}}} ∈ A^{2}(C).

This operator is unitary and the transformation T ↦ W^{∗}T W maps Z to S; hence, as before, it induces a C^{∗}-isomorphism of A(Z) = A(T_{Φ} ) onto A(S). The proof is complete. □

A variant of this result may also be obtained for A^{2}(C^{N}), N ≥ 2; however, things come off a little different this time — the corresponding C^{∗}-algebra is no longer spatially isomorphic to A(S). All the same, it is still a proper subset of B(A^{2}(C^{N})).

We shall need some results of Berger and Coburn [6]. Define

ESV := {Φ ∈ L^{∞}(C^{N}) : lim_{|z|→+∞} ess sup_{|z−w|≤1}|Φ(z) − Φ(w)| = 0}

and

BCESV := {Φ ∈ ESV : Φ is continuous on C^{N}}.

Here, as usual,

|x| = (∑_{n=1}^{N}|x_{n}|^{2})^{1/2} for x = (x_{1}, x_{2}, …, x_{N}) ∈ C^{N}.

Further, let S := {x ∈ C^{N} : |x| = 1} be the unit sphere in C^{N}.

Proposition 3.25. Let G : S → C be a continuous function on S. Define

(23)  Φ(rx) = \{ \begin{matrix}G(x) & if r ≥ 1, \\ rG(x) & if r ≤ 1, \end{matrix}  x ∈ S, 0 ≤ r < +∞.

Then

(i) Φ ∈ BCESV, and
(ii) the Hankel operators H_{Φ} , H\overline{Φ} are compact.

Assume further that

(24)  G(S) = T.

Then also (iii) σ_{e}(T_{Φ} ) = T and (iv) indT_{Φ} = 0 if N ≥ 2, and indT_{Φ} is minus the winding number of the function G : T → T (with

respect to the origin) when N = 1.

Proof. (i) Φ is continuous and bounded since G is, and Φ ∈ ESV in view of [6], Theorem 3(i).

(ii) Theorem 11 of [6] says that H_{Φ} and H\overline{Φ} are compact for arbitrary Φ ∈ ESV.

(iii) & (iv) Immediate consequences of [6], Theorem 19. □

Remark 3.26. It is possible to prove part (iv) in another way, using the idea from the end of the proof of Theorem 3.10. Suppose that θ ↦ G_{θ}, θ ∈ ⟨0, 1⟩, G_{θ} ∈ C(S), is a homotopy between G_{0} and G_{1}; construct functions Φ_{θ} according to (23) and consider the Toeplitz operators T_{Φ_{θ}} . It can be shown that T_{Φ_{θ}} are Fredholm operators ∀θ ∈ ⟨0, 1⟩, and, consequently, indT_{Φ_{0}} = indT_{Φ_{1}} . If N ≥ 2, the homotopy group π(S, T) = π_{2N−1}(T) is trivial; hence there is a homotopy connecting G_{0} = G to G_{1} = 1. It follows that

indT_{Φ} = indT_{1} = indI = 0.

If N = 1, π(T, T) = π_{1}(T) is isomorphic to Z; an isomorphism is given by G ↦ windG. It follows that there is a homotopy connecting G_{0} = G to G_{1}, G_{1}(e^{it}) = e^{kit}, k = windG, and

indT_{Φ} = indT_{Φ_{1}} = −k.

Thus the occurence of two cases — N = 1 versus N ≥ 2 — in the part (iv) is of topological nature, being related to (non)vanishing of the homotopy groups π_{n}(T). □

Theorem 3.27. Assume that the functions G : S → C and Φ : C^{N} → C satisfy the conditions (23),(24) and that N ≥ 2. Then

(a) T_{f} ∈ A(T_{Φ} ) for all f ∈ L^{∞}(C^{N}).

(b) There exists a unitary operator W : L^{2}(T) → A^{2}(C^{N}) such that the transformation T ↦ W^{∗}T W is a C^{∗}-isomorphism of A(T_{Φ} ) onto A(U), where U is the bilateral (forward) shift operator (the multiplication by z) on the Lebesgue space L^{2}(T). In particular, A(T_{Φ} ) is a C^{∗}-algebra.

(c) The operator J : L^{2}(T) → L^{2}(T), Jf(z) := f(−z), does not belong to A(U). Consequently, A(T_{Φ} ) is a proper C^{∗}-subalgebra of B(A^{2}(C^{N})).

Proof. (a) For arbitrary f ∈ L^{∞}(C^{N}),
T_{f}T_{Φ} − T_{Φ} T_{f} = H∗ ; \overline{Φ}H_{f} − H∗ ; \overline{f}H_{Φ} (cf. Proposition 1.6),

and the operators H_{Φ} , H\overline{Φ} are compact by Theorem 3.24(ii).

(b) According to Theorem 3.24(iii) & (iv), σ_{e}(T_{Φ} ) = T = σ_{e}(U) and indT_{Φ} = 0 = indU. Hence, by the Brown-Douglas-Fillmore theory [8], there exists a unitary operator W : L^{2}(T) → A^{2}(C^{N}) such that

W^{∗}T_{Φ} W = U + K,

where K ∈ Comp . The result follows in the same way as in the proof of Theorem 3.7, with S replaced by U.

(c) With respect to the standart orthonormal basis {e_{n}}_{n∈Z}, e_{n}(z) = z^{n}, z ∈ T, of L^{2}(T), the operators J and U are given by

Ue_{n} = e_{n+1} ,  Je_{n} = (−1)^{n}e_{n}  (n ∈ Z).

It follows that UJ − JU = 2UJ; but the operator UJ is unitary, and so certainly not compact. □

Remark 3.28. To be precise, we ought to check that there exist functions G and Φ satisfying the conditions (23) and (24). As an example, take G(z) = e^{4iRe z_{1}} . □

The argument above applies also in the case N = 1; one has only to replace L^{2}(T) by H^{2} and U by S^{k} or S^{∗(−k)} when k = −indT_{Φ} = windG ≠ 0. In particular, if G : T → T is the identity, we get another proof of Theorem 3.24.

What’s the relationship between A(S) and A(U)? Since H^{2} is a subspace of L^{2}(T), we may consider

B(H^{2}) to be a subset (in fact, a C^{∗}-subalgebra) of B(L^{2}(T)) by identifying T ∈ B(H^{2}) with ( \begin{matrix}T & 0 \\ 0 & 0 \end{matrix}) ∈

B(H^{2} ⊕ H_{−}^{2}), H_{−}^{2} := L^{2}(T) ⊖ H^{2}. The mapping A ↦ P_{+} A↾H^{2}, i.e.

( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) ∈ B(H^{2} ⊕ H_{−}^{2}) ⟼ A_{11} ∈ B(H^{2}),

is then a projection of B(L^{2}(T)) onto B(H^{2}).

Theorem 3.29. Under this identification, A(S) becomes A(U) ∩ B(H^{2}). Moreover, P_{+} A↾H^{2} ∈ A(S) whenever A ∈ A(U).

Proof. With respect to the decomposition L^{2}(T) = H^{2} ⊕ H_{−}^{2},

U = ( \begin{matrix}S & K_{1} \\ 0 & ∗ \end{matrix} ) ,

where K = ⟨ . , \overline{z}⟩1 is a compact operator. If T ∈ B(H^{2}), then

( \begin{matrix}T & 0 \\ 0 & 0 \end{matrix}) U − U ( \begin{matrix}T & 0 \\ 0 & 0 \end{matrix}) = ( \begin{matrix}T S & T K_{1} \\ 0 & 0 \end{matrix} ) − ( \begin{matrix}ST & 0 \\ 0 & 0 \end{matrix}) = ( \begin{matrix}[T, S] & T K_{1} \\ 0 & 0 \end{matrix} ) ,

and so ( \begin{matrix}T & 0 \\ 0 & 0 \end{matrix}) ∈ A(U) iff T ∈ A(S). As for the second assertion, let A = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) ∈ B(H^{2} ⊕ H_{−}^{2});

then

AU − UA = ( \begin{matrix}A_{11}S − SA_{11} − K_{1}A_{21} & ∗ \\ ∗ & ∗ \end{matrix} ) .

Hence A_{11} = P_{+} A↾H^{2} ∈ A(S) if A ∈ A(U), q.e.d. □

Thus, A(S) is ”smaller” then A(U) in the sense that the former is C^{∗}-isomorphic to a subalgebra of the latter. All the same, it is still possible that A(S) and A(U) are actually C^{∗}-isomorphic. We are only able to prove that, if such is the case, the isomorphism can not be spatial, i.e. of the form T ↦ W T W^{∗}, where W : H^{2} → L^{2}(T) is unitary (T ∈ A(S), W T W^{∗} ∈ A(U)). Before exhibiting the proof, let us first establish some general properties of the algebras A.

In general, there are two candidates for the definition of A, which have happened to coincide in all cases encountered so far. Let H be a separable infinite-dimensional Hilbert space and M ∈ B(H). Define

A(M) := {T ∈ B(H) : MT − T M ∈ Comp (H)}

and

A^{♯}(M) := {T ∈ B(H) : T − M^{∗}T M ∈ Comp (H)}.

The first investigation of A^{♯}(S) is reported to have been done by Barría and Halmos, whose results, unfortunately, appeared only as a rather unavailable preprint [4], and so remain unknown to the present author. Afterwards these spaces seem to have received almost no attention at all, although many results on essential commutants may be phrased in terms of them. Let us mention the theorem of Johnson and Parrot [20] which says that

⋂_{ϕ∈L^{∞}(T)} A(M_{ϕ}) = {M_{ϕ} + K : ϕ ∈ L^{∞}(T), K ∈ Comp },

where M_{ϕ} is the operator of multiplication by ϕ on L^{2}(T), and two results of Davidson [11] concerning Toeplitz operators on H^{2}:

⋂_{θ inner} A^{♯}(T_{θ}) = {T_{ϕ} + K : ϕ ∈ L^{∞}(T), K compact },

and

⋂_{θ inner} (A(T_{θ}) ∩ A(T_{θ}^{∗})) = {T_{ϕ} + K : ϕ ∈ QC, K compact}.

(QC = (H^{∞} + C) ∩ \overline{H^{∞} + C} are the quasicontinuous functions on T.)

The following proposition describes elementary properties of A(M) and A^{♯}(M).

Proposition 3.30. (a) A(M) and A^{♯}(M) are norm-closed subspaces of B(H).

(b) A(M) is an operator algebra with identity. It contains M, Comp , and may be even all of B(H). It is a C^{∗}-algebra if M is selfadjoint; in general, A(M^{∗}) = A(M)^{∗}.

(c) A^{♯}(M) is a selfadjoint set containing Comp . It may be all of B(H), but may also reduce merely to Comp .

(d) A^{♯}(M) is C^{∗}-algebra if I −MM^{∗} ∈ Comp , and contains the identity if and only if I −M^{∗}M ∈ Comp .

(e) A(M) = A^{♯}(M) ⟺ M is essentially unitary, and then it is a C^{∗}-algebra with identity.

Proof. (a) Obvious.

(b) If A, B ∈ A(M), then [AB, M] = A[B, M] + [A, M]B is also compact, hence AB ∈ A(M). The assertions I, M ∈ A(M), Comp⊂ A(M) are immediate, and so is A(M^{∗}) = A(M)^{∗}. If M is selfadjoint, M^{∗} = M, whence A(M) = A(M)^{∗} is also selfadjoint, and so is a C^{∗}-algebra. If M ∈ Comp , A(M) = B(H).

(c) The first two assertions are straightforward. If M ∈ Comp , A^{♯}(M) = Comp ; if M = I, A^{♯}(M) = B(H).

(d) The second statement is clear; if I − MM^{∗} ∈ Compand A, B ∈ A^{♯}(M), then

AB − M^{∗}ABM = A(B − M^{∗}BM) + (A − M^{∗}AM)M^{∗}BM − M^{∗}A(I − MM^{∗})BM ∈ Comp ,

and so AB ∈ A^{♯}(M). Owing to (a) and (b), A^{♯}(M) must be a *-subalgebra of B(H), hence a C^{∗}-algebra.

(e) If A(M) = A^{♯}(M), then I ∈ A^{♯}(M) owing to (b), and so I − M^{∗}M ∈ Comp ; moreover, M ∈ A(M) ⇒M ∈ A^{♯}(M) ⇒M^{∗} ∈ A^{♯}(M) ⇒M^{∗} ∈ A(M) owing to (c), i.e. M is essentially normal. It follows that M is essentially unitary. On the other hand, if M is essentially unitary, then

A − M^{∗}AM ∈ Comp⟹ M(A − M^{∗}AM) = MA − AM + (I − MM^{∗})AM ∈ Comp⟹
⟹ MA − AM ∈ Comp ,

and

MA − AM ∈ Comp⟹ M^{∗}(MA − AM) = A − M^{∗}AM − (I − M^{∗}M)A ∈ Comp⟹
⟹ A − M^{∗}AM ∈ Comp ,

hence A(M) = A^{♯}(M); it is a C^{∗}-algebra by virtue of (d). □

The last two properties suggest it might be helpful to try to describe A(M) and A^{♯}(M) in terms of the image m of M in the Calkin algebra

Calk(H) := B(H)/Comp (H).

We will denote the images of R, S, T, M, … ∈ B(H) in Calk(H) by corresponding small letters r, s, t, m, …, and by π the canonical projection B(H) → Calk(H); π is a C^{∗}-homomorphism.

Proposition 3.31. (a) T ∈ A(M) ⟺ t ∈ π(A(M)), and similarly for A^{♯}(M).

(b) T ∈ A(M) ⟺ t ∈ m′, the commutant of m in Calk(H).

(c) T ∈ A^{♯}(M) ⟺ t = m^{∗}tm.

(d) A(M) is a C^{∗}-algebra iff m is a normal element of Calk(H).

(e) m′′ ⊂ m′; in particular, continuous functions of m belong to π(A(M)).

(f) A(M) ⊂ A(R) iff r ∈ m′′.

Proof. (a) A direct consequence of the fact that A(M), resp. A^{♯}(M) contain Comp .

(b) and (c) Straightforward.

(d) A(M) is a C^{∗}-algebra iff m′ is. In that case, m^{∗} ∈ m′, and so m is normal. On the other hand, if m is normal and tm = mt, then t^{∗}m = mt^{∗} by the Putnam-Fuglede theorem; hence t^{∗} ∈ m′, and so m′ is a C^{∗}-algebra.

(e) m ∈ m′ implies m′′ ⊂ m′; continuous functions of m belong to m′′; π(A(M)) = m′ by (a).

(f) A(M) ⊂ A(R) iff m′ ⊂ r′. If this is the case, then m′′ ⊃ r′′ and so r ∈ r′′ ⊂ m′′. On the other hand, if r ∈ m′′, then r′ ⊃ m′′′ = m′. □

As a parting shot, we prove the promised theorem that the algebras A(U) and A(S) are not spatially isomorphic. A result of Johnson and Parrot, already alluded to above, will be needed:

Theorem 3.32. If T is an operator on L^{2}(T) such that [T, M] is compact for all operators M of the form

(25)  M = M_{ϕ} + K,  ϕ ∈ L^{∞}(T), K compact,

then T is itself of this form.

Proof. See [20]. □

The Johnson-Parrot theorem may be rephrased in the following way: Let

L_{∞} := {π(M_{ϕ}) : ϕ ∈ L^{∞}(T)} ⊂ Calk(L^{2}(T)).

Then L_{∞} coincides with its commutant: L_{∞}′ = L_{∞}.

Lemma 3.33. If an operator M of the form (25) is Fredholm, then indF = 0.

Proof. We shall use the previous theorem^{7} ^{8}. If M = M_{ϕ} + K_{1} is Fredholm, then m ≡ π(M) ∈ Calk is invertible. If t ∈ L_{∞}, then mt = tm and so

0 = m^{−1}(mt − tm)m^{−1} = tm^{−1} − m^{−1}t,

whence m^{−1} ∈ L_{∞}′. By virtue of the Johnson-Parrot theorem, m^{−1} ∈ L_{∞}; consequently, there exist ψ ∈ L^{∞}(T) and K_{2} ∈ Compsuch that

(M_{ϕ} + K_{1}).M_{ψ} = I + K_{2},

which implies that

M_{ϕψ−1} ∈ Comp .

Let {z^{n}}_{n∈Z} be the standard basis of L^{2}(T); since z^{n}→^{w}0, the last condition forces

∥M_{ϕψ−1}z^{n}∥ → 0.

But ∥M_{ϕψ−1}z^{n}∥^{2} = ∥ϕψ − 1∥_{2}^{2} for all n ∈ Z; therefore ϕψ = 1 and M_{ψ} is the inverse of M_{ϕ}. It follows that indM_{ϕ} = 0, and so likewise indM = ind(M_{ϕ} + K_{1}) = 0. □

Theorem 3.34. Let S be the unilateral forward shift on H^{2}, U the bilateral forward shift on L^{2}(T), W : H^{2} → L^{2}(T) a unitary operator, and Y = W SW^{∗}. Then it cannot happen that A(U) ⊂ A(Y). In particular, the algebras A(U) and A(Y) are not spatially isomorphic.

Proof. By virtue of the part (f) of the last proposition, A(U) ⊂ A(Y) is equivalent to y ∈ u′′. Since U = M_{z} is of the form (25), we have u ∈ L_{∞} and the Johnson-Parrot theorem implies that

u′′ ⊂ L_{∞}′′ = L_{∞} = L_{∞}′ ⊂ u′.

Hence y ∈ L_{∞}, i.e. Y = M_{ϕ} + K for some ϕ ∈ L^{∞}(T) and K ∈ Comp . Because Y = W SW^{∗},

σ_{e}(Y) = σ_{e}(S) = T,

so Y is Fredholm. In view of the preceding lemma, indY = 0. But

indY = ind(W SW^{∗}) = indS = −1

— a contradiction. Thus it can never happen that A(U) ⊂ A(Y), or even A(U) = A(Y). □ We close this chapter with two open problems. Problem. Are the C^{∗}-algebras A(U) and A(S) (non-spatially) isomorphic? Problem. Denote W_{B} and W_{F} the unitary operators

W_{B} : H^{2} → A^{2}(D), z^{n} ∈ H^{2} ↦ \sqrt{n + 1} z^{n} ∈ A^{2}(D),

7A more elementary proof is bound to exist, but we won’t bother with it, since we are going to use the Johnson-Parrot theorem in the sequel anyway. 8O.K., here is an elementary proof. Since F, hence also M_{ϕ}, is Fredholm, kerM_{ϕ} must be finite-dimensional and RanM_{ϕ} must be closed. If ϕ vanished on a set E ⊂ T of positive measure, the kernel of M_{ϕ} would contain an isometric copy of L^{2}(E) and thus would not be finite-dimensional. Consequently, ϕ ≠ 0 almost everywhere on T; it follows that M_{ϕ} is injective. By the open mapping theorem, RanM_{ϕ} is closed if and only if M_{ϕ} is bounded below, which takes place iff |ϕ| is bounded away from zero. But then M_{ϕ} is invertible, its inverse being M_{1/ϕ}; hence, indM_{ϕ} = 0, and so indF = 0 as well. resp.

W_{F} : H^{2} → A^{2}(C), z^{n} ∈ H^{2} ↦ (n! 2^{n})^{−1/2}z^{n} ∈ A^{2}(C),

constructed in the paragraph following Corollary 3.6 and in course of the proof of Theorem 3.24(iii), respectively. Denote further

B_{ϕ} := W_{B}^{∗}T_{ϕ}W_{B} for ϕ ∈ L^{∞}(D),

resp.

F_{ϕ} := W_{F}^{∗}T_{ϕ}W_{F} for ϕ ∈ L^{∞}(C),

and let T_{B}, T_{F} be the C^{∗}-subalgebras of B(H^{2}) generated by {B_{ϕ} : ϕ ∈ L^{∞}(D)} and {F_{ϕ} : ϕ ∈ L^{∞}(C)}, respectively. Theorems 3.2 and 3.24 then assert that

T_{B} ⊂ A(S) and T_{F} ⊂ A(S).

Are these inclusions strict?

Chapter 4. THE TOEPLITZ CALCULUS

In this chapter, Toeplitz operators both on the Hardy space and the Bergman space, as well as on the Fock space, are dealt with; to prevent confusion, the following convention will be observed throughout: T_{ϕ} always denotes a Toeplitz operator on H^{2} (i.e. ϕ ∈ L^{∞}(T)), B_{ϕ} stands for a Toeplitz operator on A^{2}(D) (ϕ ∈ L^{∞}(D)), and F_{ϕ} for a Toeplitz operator on the Fock space A^{2}(C) (ϕ ∈ L^{∞}(C)). The algebra A(S) will frequently be abbreviated to A. When convenient, the spaces A^{2}(D) and A^{2}(C) will be identified with H^{2} by means of the unitary operators W_{B} and W_{F} , respectively, mentioned at the very end of the previous chapter.

We begin by recalling several facts from the theory of Toeplitz operators on H^{2}. Denote

τ_{H} := {T_{ϕ} : ϕ ∈ L^{∞}(T)}

and let T_{H} be the C^{∗}-algebra generated by τ_{H}. Further, define

(26)  𝔖_{0} := {T ∈ B(H^{2}) : lim_{n→∞} ∥T e_{n}∥ = 0} = {T ∈ B(H^{2}) : lim_{n→∞} ∥T S^{n}f∥ = 0 ∀f ∈ H^{2}},

where S = T_{z} is the (forward) shift operator with respect to the standard basis {e_{n}}_{n∈N} , e_{n}(z) = z^{n}, of H^{2}. The equality (26) is easily verified: if T e_{n} → 0, then T S^{n}p → 0 for every polynomial p; since these are dense in H^{2} and the operators T S^{n} are uniformly bounded (by ∥T ∥), T S^{n}f → 0 for any f ∈ H^{2}.

The following are classical results from the theory of Toeplitz operators on H^{2}. The abelianized algebra T_{H}, i.e. T_{H} factored by its commutator ideal ComT_{H}, is (isometrically) C^{∗}-isomorphic to L^{∞}(T), i.e. there is a contractive map

(27)  ξ : T_{H} → L^{∞}(T)

which is linear, multiplicative, surjective, preserves adjoints, and kerξ = ComT_{H}. The mapping

ϕ ↦ T_{ϕ}

is an isometric cross-section of ξ, i.e. ξ(T_{ϕ}) = ϕ. Thus, every operator T ∈ T_{H} admits a unique decomposition

T = T_{ϕ} + X, X ∈ ComT_{H}, ϕ ∈ L^{∞}(T),

and ϕ = ξ(T); that’s why the mapping ξ is sometimes called the symbol map. This map, in fact, can be described somewhat more explicitly: it can be shown that

ComT_{H} ⊂ 𝔖_{0};

consequently, for arbitrary T ∈ T_{H}, the limit

Mf := lim_{n→∞} U^{∗n}T S^{n}f

exists for every f ∈ H^{2}. Here H^{2} is thought of as a subspace of L^{2}(T) and U is the (bilateral, forward) shift operator on L^{2}(T). The resulting operator M : H^{2} → L^{2}(T) is bounded (by ∥T ∥, as a matter of fact) and commutes with U, so it must be of the form M_{ϕ} for some ϕ ∈ L^{∞}(T). Now it’s already easy to verify that ϕ = ξ(T). Thus

ξ(T) = lim_{n→∞} U^{∗n}T e_{n} ∈ L^{∞}(T) ⊂ L^{2}(T).

We are not going to prove the facts mentioned above, but rather refer to the books of Douglas [12] or Nikolskiǐ[22].

Note also that, owing to the multiplicativity of ξ,

(28)  ξ(T_{ϕ}T_{ψ}) = ξ(T_{ϕ})ξ(T_{ψ}) = ϕψ = ξ(T_{ϕψ})  ∀ϕ, ψ ∈ L^{∞}(T),

which implies that kerξ = ComT_{H} contains not only the commutator [T_{ϕ}, T_{ψ}], but even the semi-commutator

[T_{ϕ}, T_{ψ}) := T_{ϕψ} − T_{ϕ}T_{ψ}

of T_{ϕ} and T_{ψ}.

The objective of this chapter is an attempt to construct a similar theory, or ”symbol calculus”, for the algebras T_{B}, T_{F} or even A. (The algebra A(U) shall not be explicitely considered, the reason being that the larger the algebra, the less chance to construct some calculus; and A(U) is ”larger” than A(S) ≡ A — cf. Theorem 3.29. All the same, most of the ideas below can be applied to A(U) as well.)

The most naive idea (e.g. for T_{B}) is to try to obtain a continuous linear mapping

(29)  ξ_{B} : T_{B} → L^{∞}(D)

which is be multiplicative and admits ϕ ↦ B_{ϕ} as a cross-section. Unfortunately, this is doomed to a failure: the following lemma shows that ComT_{B} ⊃ Comp , and so if ξ_{B} is to be multiplicative, its kernel must contain Comp . But there exist non-zero compact Toeplitz operators on A^{2}(D) (cf. Corollary 1.5), and so ϕ ↦ B_{ϕ} cannot be a cross-section of ξ_{B}. For T_{F} and A the same situation occurs.

Lemma 4.1. (a) The algebras T_{B}, T_{F} and A contain Comp .

(b) The commutator ideals ComT_{B}, ComT_{F} and ComA contain Comp .

Proof. (a) T_{B} and T_{F} contain Compin view of Theorem 2.4, and A ⊃ T_{B}.

(b) The algebras T_{B}, T_{F} and A are not commutative, so it suffices to show that Compis the smallest proper (closed, two-sided, selfadjoint) ideal in each of them. We shall prove this for T_{B}; the proof for T_{F} is similar, and ComA ⊃ ComT_{B} since A ⊃ T_{B}.

So let I be a proper (closed, two-sided, ∗– ) ideal in T_{B}. Take a nonzero T ∈ I and vectors x, y ≠ 0 such that T x = y. In view of (a), the operators A = ⟨ . , e⟩x and B = ⟨ . , y⟩f belong to T_{B} for arbitrary e, f. Since I is an ideal, BT A = ∥y∥^{2}⟨ . , e⟩f belongs to I. It follows that I contains all rank one operators, hence all finite rank operators (by linearity), which are dense in Comp . □

So let us give up multiplicativity, and ask only for a continuous linear map (29) which admits ϕ ↦ B_{ϕ} as a cross-section. Unfortunately, we are doomed to fail once again. The reason is that ξ_{B}(B_{ϕ}) = ϕ implies ∥B_{ϕ}∥ ≥ ∥ξ_{B}∥^{−1}∥ϕ∥_{∞}, i.e. the mapping ϕ ↦ B_{ϕ} would have to be bounded below. This is, however, easily seen not to be the case. If χ is the characteristic function of the disc {z : |z|^{2} ≤ R}, 0 < R < 1, then B_{ϕ} is a diagonal operator with weights (cf. Proposition 1.3)

c_{n} = ∫_{0}^{R}(n + 1)t^{n} dt = \frac{R^{n+1}}{n + 1} ,

which implies ∥B_{ϕ}∥ = R while ∥ϕ∥_{∞} = 1.

This suggests that, perhaps, there is nothing wrong with multiplicativity, but rather with the target space L^{∞}(D); so let us insist on multiplicativity and try to replace L^{∞}(D) by something else. A natural candidate for the ”something” turns up quickly. Namely, the C^{∗}-algebras A/ComA, T_{B}/ComT_{B} and T_{F} /ComT_{F} are commutative, and therefore C^{∗}-isomorphic to the spaces of all continuous functions on their maximal ideal spaces, via the Gelfand transform. It remains to describe these spaces and see whether they are not somehow connected with L^{∞}(D) (or L^{∞}(C), respectively) — for instance, they might be homeomorphic to some subsets of the maximal ideal space of L^{∞}(D) (or of L^{∞}(C), for that matter — these two are homeomorphic) , or something like that. This description in turn amounts to identifying the multiplicative linear functionals on A/ComA, T_{B}/ComT_{B} and T_{F} /ComT_{F} , respectively — or, which is the same, on A, T_{B}, T_{F}. Thus we are lead to a fundamental question: are there any multiplicative linear functionals on these algebras at all?

Before going on, let us make two remarks. First, one might try to obtain a multiplicative linear functional e.g. on A by restricting to A a multiplicative linear functional on B(H^{2}). This is, however, impossible — there are no multiplicative linear functionals on B(H^{2}) and, consequently, ComB(H^{2}) = B(H^{2}). To see this, decompose H^{2} into H_{1} ⊕ H_{2}, where both H_{1} and H_{2} are infinite-dimensional. Then there exists a unitary operator which maps H_{1} onto H_{2} and vice versa; denote it V, and let P_{i} stand for the orthogonal projection onto H_{i} (i = 1, 2). Now, if ϕ were a nontrivial multiplicative linear functional on B(H^{2}),

P_{1} + P_{2} = I ⟹ ϕ(P_{1}) + ϕ(P_{2}) = 1,
V^{∗}V = I ⟹ \overline{ϕ(V)}ϕ(V) = 1,
V^{∗}P_{1}V = P_{2} ⟹ ϕ(P_{2}) = ϕ(V^{∗})ϕ(P_{1})ϕ(V) = ϕ(P_{1}),

so ϕ(P_{1}) = ϕ(P_{2}) = 1/2; but

P_{1}P_{2} = 0 ⟹ \frac{1}{4} = ϕ(P_{1})ϕ(P_{2}) = 0,

a contradiction.

Second, we have noted that the symbol map ξ for Toeplitz operators on H^{2} satisfies (28)

ξ(T_{ϕ}T_{ψ} − T_{ϕψ}) = 0.

It should be pointed out that this relation cannot be satisfied on the Bergman space A^{2}(D) or the Fock space A^{2}(C), i.e. for algebras T_{B} or T_{F} (or A): the following example shows that there even exists ϕ ∈ L^{∞}(D) such that [B_{ϕ}^{∗}, B_{ϕ}) is a nonzero multiple of the identity.

Example 4.2. Define the function ϕ ∈ L^{∞}(D) by

ϕ(z) = exp(i lnln \frac{1}{|z|^{2}} ).

In view of Proposition 1.3, B_{ϕ} is a diagonal operator with weights

c_{n} = ∫_{0}^{1} exp(i lnln \frac{1}{t}) . (n + 1)t^{n} dt.

Changing the variable to w = z^{n+1} gives

c_{n} = ∫_{0}^{1} exp(i lnln \frac{1}{w} − i ln(n + 1)) dw = e^{−i ln(n+1)} ∫_{0}^{1} exp(i lnln \frac{1}{w}) dw.

Substituting once more, namely, y = ln \frac{1}{w} , yields

∫_{0}^{1} exp(i lnln \frac{1}{w}) dw = ∫_{0}^{+∞} exp(i lny) . e^{−y} dy = Γ(i + 1) (popularly ”i!”).

Also, B\overline{ϕ} = B_{ϕ}^{∗} = diag(\overline{c_{n}}) and B_{\overline{ϕ}ϕ} = I since ϕ is unimodular on D. Summing up, we have

B_{\overline{ϕ}ϕ} − B\overline{ϕ}B_{ϕ} = diag(1 − |c_{n}|^{2}) = diag(1 − |Γ(i + 1)|^{2}) = (1 − |Γ(i + 1)|^{2}).I

It remains to show that |Γ(i + 1)| ≠ 1. To that aim, recall the formulas for the gamma function

Γ(x + 1) = xΓ(x),  Γ(\overline{x}) = \overline{Γ(x)},  Γ(x)Γ(1 − x) = \frac{π}{sinπx},

and compute:

|Γ(i + 1)|^{2} = Γ(i + 1)Γ(1 − i) = iΓ(i)Γ(1 − i) =

= \frac{πi}{sinπi} = \frac{2π}{e^{π} − e^{−π}} = 0, 272 … < 1.

On the Fock space, a similar counterexample can be constructed. The function ϕ ∈ L^{∞}(C),

ϕ(z) = exp(i|z|^{2}/2),

is unimodular; the corresponding Toeplitz operator F_{ϕ} is diagonal (cf. Proposition 1.7) with weights

c_{n} = ∫_{0}^{+∞} e^{it} . \frac{t^{n}e^{−t}}{n!} dt.

Expanding e^{it} into the Taylor series and integrating term by term^{9} yields

c_{n} = ∑_{k=0}^{∞} \binom{n + k}{k}i^{k} = (1 − i)^{−n−1}.

Consequently, F_{\overline{ϕ}ϕ} −F\overline{ϕ}F_{ϕ} = diag(1−|c_{n}|^{2}) = diag(1−2^{−n−1}), which is an invertible operator and so cannot belong to kerξ_{F} for any nonzero ξ_{F}. □

Now let us come back to the search of multiplicative linear functionals on the three algebras T_{B}, T_{F} and A. Owing to Lemma 4.1, every multiplicative linear functional on A must vanish on Comp ; consequently, it suffices to look for multiplicative linear functionals on A/Comp , and similarly for T_{B} and T_{F}. These are subalgebras of the Calkin algebra Calk(H^{2}) = B(H^{2})/Comp (H^{2}), which suggests that some techniques used for the study of Calk might turn useful. The algebra A/Comp⊂ Calk admits a particularly easy description (cf. Proposition 3.31(b)): it coincides with the commutant of the image s = π(S) of the shift operator S in Calk(H^{2}). For this reason, we shall be concerned mainly with A from now on, but most of what is said applies to T_{B} and T_{F} as well.

In the classical (i.e. H^{2}) case, a powerful tool is provided by the dilation theory, which enables one to ”lift”, in some sense, a Toeplitz operator T_{ϕ} from H^{2} to an operator on a larger space (namely, L^{2}(T)) enjoying certain properties (see e.g. [25]). It turns out that a similar lifting can be constructed using certain representation of the Calkin algebra. The classical lifting mentioned above (i.e. of T_{ϕ} to L^{2}(T)) may then be obtained by passing to certain subspace, isomorphic to L^{2}(T) (the subspace is the same for all operators T_{ϕ}). We proceed to describe the representation of Calk(H^{2}); the idea goes back to Calkin [10].

Denote by π the canonical projection of B(H^{2}) onto Calk(H^{2}); to simplify the notation, we shall often write (once again) t, s, etc. for π(T), π(S), etc.

We begin by a brief excursion into Banach limits. Loosely speaking, they are extensions of the usual ”lim” to the space l^{∞} of all bounded sequences of complex numbers. In more precise terms, a Banach limit is a mapping Lim from l^{∞} into C which satisfies the following conditions:

(C1) it is linear, of norm 1, and extends the usual lim (i.e. Limf_{n} = limf_{n} whenever the latter exists); (C2) it preserves complex conjugation, i.e. Lim \overline{f_{n}} = \overline{Limf_{n}} ∀\{f_{n}\} ∈ l^{∞}); consequently, Limf_{n} is a

real number when f_{n} is real for all n ∈ N; (C3) it is positive, i.e. f_{n} ≥ 0 ∀n ∈ N implies Limf_{n} ≥ 0.

To obtain such a functional, one may procced as follows. First, use the Hahn-Banach theorem to extend ”lim” to a linear functional on l^{∞} without increasing the norm; this gives a functional ”Lim” which satisfies

(C1). Next, replace Lim by Lim \frac{f_{n} + \overline{f_{n}}}{2} if necessary; the resulting functional satisfies both (C1) and (C2). Finally, observe that (C1) and (C2) already imply (C3). It suffices to show that if 0 ≤ f_{n} ≤ 1 for all n, then 0 ≤ Limf_{n} ≤ 1. But

(0 ≤ f_{n} ≤ 1 ∀n) ⟺ (f_{n} is real, |f_{n}| ≤ 1 and |1 − f_{n}| ≤ 1 ∀n) ⟹
⟹ (Limf_{n} is real, | Limf_{n}| ≤ 1 and |1 − Limf_{n}| ≤ 1) ⟺
(by (C1) and (C2)) ⟺ 0 ≤ Limf_{n} ≤ 1.

9This is legal for ∫_{0}^{1}, since the Taylor series converges uniformly for 0 < t < 1; as for ∫_{1}^{+∞} , split the series into four (corresponding to i^{k} = ±1, ±i) and apply the Lebesgue Monotone Convergencs Theorem to each.

Thus, Banach limits exist in abundance. There are two additional conditions one could impose: (C4) Limf_{n+1} = Limf_{n}; (C5) Limf_{n}g_{n} = Limf_{n} Limg_{n}.

An unpleasant thing is that these two conditions conflict with each other: let

x_{n} = \{ \begin{matrix}1 & if n is even \\ 0 & if n is odd \end{matrix} , y_{n} = x_{n+1} .

If there were a ”Lim” which satisfies both (C4) and (C5), we would have Limx_{n} = Limy_{n} (by (C4)) and Lim(x_{n} + y_{n}) = 1 , whence Limx_{n} = Limy_{n} = 1/2; but Limx_{n} . Limy_{n} = 0 by (C5) — a contradiction.

All the same, it is quite easy to construct a Banach limit which satisfies either only (C4), or only (C5). The former may be obtained from every Banach limit by replacing it with Lim(\frac{1}{n+1} ∑_{k=0}^{n} x_{k}). This does not affect (C1) — (C3) and

Lim(\frac{1}{n + 1} ∑_{k=0}^{n}x_{k}) − Lim(\frac{1}{n + 1} ∑_{k=0}^{n}x_{k+1} ) = Lim \frac{x_{0} − x_{n+1}}{n + 1} = 0.

To obtain a Banach limit satisfying (C5), adopt a different approach. With pointwise multiplication, l^{∞} is a commutative Banach algebra, and hence is isomorphic, by means of the Gelfand transform, to the space C(𝔐) of all continuous (complex-valued) functions on its maximal ideal space 𝔐. This space is usually denoted βN, since it coincides with the Stone-Čech compactification of the set N equipped with the discrete topology. The elements of 𝔐 = βN are multiplicative linear functionals on N and are known to satisfy (C1)

— (C3) and (C5), except for the fact that they need not extend the functional ”lim”. The set N can be embedded into βN in a natural way; the multiplicative linear functional corresponding to n ∈ N is given by

n̂ : {x_{k}} ∈ l^{∞} ↦ x_{n} ∈ C.

We assert that if ϕ ∈ βN \ N, then ϕ extends ”lim”. To prove this, it suffices to check that ϕ({x_{n}}) = 0 whenever lim_{n→∞} x_{n} = 0. Since N is dense in βN, there exists a net {n_{ι}}_{ι∈Λ} ⊂ N such that n̂_{ι} → ϕ. If there were a number m ∈ N which occured in {n_{ι}}_{ι∈Λ} an infinite number of times, we would have ϕ = m, contrary to our assumption ϕ ∈ βN \ N. It follows that n_{ι} → ∞, whence

ϕ({x_{k}}) = lim_{ι∈Λ} n̂_{ι}({x_{k}}) = lim_{ι∈Λ}x_{n_{ι}} = 0

whenever lim_{k→∞} x_{k} = 0, and our claim is verified.

Thus, Banach limits satisfying either (C4) or (C5) exist in abundance, too. In the sequel, a Banach limit, always denoted ”Lim”, is assumed to satisfy only (C1) — (C3); it will be pointed out explicitely when one of the additional properties is required.

Now we may proceed to introduce the Calkin representation. Let

L′′ := { {x_{k}}_{k∈N} : x_{k} ∈ H^{2} ∀k ∈ N, and x_{k}→^{w}0}

be the set of all sequences of elements of H^{2} which tend weakly to zero. With componentwise addition and scalar multiplication, L′′ becomes a vector space. Let Lim be a Banach limit. The functional^{10}

∥x∥_{L} := (Lim∥x_{k}∥_{H^{2}}^{2})^{1/2}, x = {x_{k}} ∈ L′′

is easily seen to be a pseudonorm on L′′. The factor space

L′ := L′′/{x ∈ L′′ : ∥x∥_{L} = 0}

10If Lim satisfies (C5), ∥x∥_{L} equals Lim∥x_{k}∥. is a normed linear space with respect to the norm ∥x∥_{L}; in fact, it is even a pre-Hilbert space with respect to the scalar product

⟨x, y⟩_{L} := Lim⟨x_{n}, y_{n}⟩, x, y ∈ L′′.

Let L be the completion of L′. Then L is a Hilbert space and L′ is a dense subset of L. For notational convenience, the elements of L′ will be denoted simply as x = {x_{n}}, y = {y_{k}}, etc., which is not so cumbersome as writing rigorously x mod{w ∈ L′′ : ∥w∥_{L} = 0}, or something similar.

Remark 4.3. The space L is uncomfortably large — it is not separable. This can be seen as follows. Let {q_{n}}_{n∈N} be the set of all rational numbers (q_{m} ≠ q_{k} if m ≠ k). For each irrational number α, select a sequence Λ_{α} ⊂ N such that lim_{j∈Λ_{α}} q_{j} = α. Thus if α ≠ β are irrational numbers, the sequences Λ_{α} and Λ_{β} have at most a finite number of terms in common. Let X_{α} be the sequence {e_{j}}_{j∈Λ_{α}} , where {e_{j}}_{j∈N} is, as before, the standard orthonormal basis of H^{2}. Then X_{α} ∈ L′′ and

∥X_{α} − X_{β}∥_{L}^{2} = Lim∥e_{Λ_{α}(j)} − e_{Λ_{β}(j)} ∥^{2} = 2

for arbitrary irrational numbers α ≠ β. Since there are uncountably many irrational numbers, L must be inseparable. □

Let T be a bounded linear operator on H^{2}. If x_{n}→^{w}0 in H^{2}, then also T x_{n}→^{w}0. Consequently, the mapping

{x_{k}} ↦ {T x_{k}}

maps L′′ into L′′; because

Lim∥T x_{k}∥^{2} ≤ Lim(∥T ∥^{2} . ∥x_{k}∥^{2}) = ∥T ∥^{2} . Lim∥x_{k}∥^{2},

it does not increase norms, and so can be extended to a bounded linear operator on L, which will be denoted T^{♯}. Thus T^{♯} is a linear operator on L and

∥T^{♯}∥_{L→L} ≤ ∥T ∥.

Let us establish some properties of the space L and the transformation T ↦ T^{♯} which will be needed in the sequel.

Proposition 4.4. The transformation T ↦ T^{♯} is linear and contractive, (T^{∗})^{♯} = (T^{♯})^{∗} and (T_{1}T_{2})^{♯} =

T_{1}^{♯}T_{2}^{♯}.

Proof. The first two assertions have been verified in the preceding paragraph; as for the other two, it suffices to check them on elements of L′′, which reduces to direct consequences of the properties of Banach limits. □

Proposition 4.5. T^{♯} = 0 iff T ∈ Comp .

Proof. The last proposition implies that the kernel of the transformation T ↦ T^{♯} is an ideal (twosided, closed, selfadjoint) in B(H^{2}). There are only three such ideals : {0}, Compand the whole B(H^{2}). Since (I_{H^{2}})^{♯} = I_{L}, it suffices to show that K^{♯} = 0 if K ∈ Comp . But K ∈ Compand x_{n}→^{w}0 implies ∥Kx_{n}∥ → 0, i.e. K^{♯}x = 0 for x ∈ L′′; by continuity, K^{♯} = 0. □

Corollary 4.6. ∥T^{♯}∥ ≤ ∥T ∥_{e}, the essential norm of T. Consequently, the transformation T ↦ T^{♯} induces a mapping t ↦ t^{♯} from the Calkin algebra Calk(H^{2}) into B(L). This mapping is an (isometric) C^{∗}-isomorphism of Calk onto a C^{∗}-subalgebra of B(L). Hence, we have even ∥T^{♯}∥ = ∥T ∥_{e}.

Proof. The preceding two propositions imply that

∥T^{♯}∥ = ∥T^{♯} + K^{♯}∥ ≤ ∥T + K∥

for every compact operator K; hence ∥T^{♯}∥ ≤ ∥T ∥_{e}. The rest follows from Propositions 4.4 & 4.5 and the fact that an injective C^{∗}-homomorphism is isometric. □

Corollary 4.7. T ∈ A ⟺ t ∈ s′ ⟺ T^{♯} ∈ S^{♯′}.

Proof. Immediate from the last corollary. □

Proposition 4.8. (1) The mapping

J : x ∈ H^{2} ↦ {S^{n}x}_{n∈N} ∈ L′′

is an isometrical isomorphism of H^{2} onto a subspace H ⊂ L. An orthonormal basis for H is given by

(30)  E_{n} = Je_{n} = S^{♯n}E_{0} = {e_{n+k} }_{k∈N} ∈ L′′,  n ∈ N,

where {e_{k}}_{k∈N} is the standard orthonormal basis of H^{2}.

(2) The operator S^{♯} behaves on H as S on H^{2}: H is an invariant subspace for S^{♯} and S^{♯}J = JS, i.e. S^{♯}E_{N} = E_{N+1} .

(3) The operator S^{♯} ∈ B(L) is unitary.

Proof. (1) Since S^{∗k} → 0 in SOT, S^{k}x→^{w}0 for every x ∈ H^{2}, so {S^{k}x}_{k∈N} ≡ Jx ∈ L′′. Further,

∥Jx∥_{L}^{2} = Lim∥S^{k}x∥^{2} = Lim∥x∥^{2} = ∥x∥^{2},

i.e. J is an isometry. Since {e_{n}} is an orthonormal basis for H^{2}, {Je_{n}} must be an orthonormal basis for H = RanJ; the formulas (30) are immediate.

(2) It suffices to check that S^{♯}E_{N} = E_{N+1} , and that’s immediate from the definitions.

(3) S^{∗}S = I, hence S^{♯∗}S^{♯} = I_{L}; I −SS^{∗} is a compact operator (namely, ⟨ . , e_{0}⟩e_{0}), hence I_{L} −S^{♯}S^{♯∗} = 0. So, indeed, S^{♯} is unitary. □

The last Proposition has an interesting corollary. The shift operator S on H^{2} has a minimal unitary dilation (in fact, a minimal unitary extension) sensu Nagy-Foias; the latter, in fact, can be identified with the bilateral shift operator U = M_{z} on L^{2}(T). Owing to parts (2) and (3) of the last Proposition, S^{♯} is a unitary extension (hence, of course, also a unitary dilation) of the operator S^{♯}↾H which is unitarily equivalent to the operator S on H^{2}. It follows that S^{♯} must contain the minimal unitary dilation U = M_{z} of S — there has to be a subspace K of L which contains H, is invariant under S^{♯}, and the restriction S^{♯}↾K is unitarily equivalent to M_{z} on L^{2}(T). This subspace can be, in fact, described explicitly: it is K = ⋁_{n∈Z} S^{♯n}E_{0}. The

vectors E_{N} := S^{♯N}E_{0}, N ∈ Z, form an orthonormal basis for K. These vectors belong to L′′, and so may be written down explicitely:

E_{N} = \{ \begin{matrix}\{ \underbrace{0, …, 0}_{−N zeroes} , e_{0}, e_{1}, e_{2}, …\} & if N < 0, \\ \{e_{N}, e_{N+1} , e_{N+2} , …\} & if N ≥ 0 (as in (30) ). \end{matrix}

Both expressions may be written as E_{N} = {S^{k+N} e_{0}}_{k∈N} if we agree to let the undefined terms (for N < 0, 0 ≤ k < −N) be zero. The isometry J : H^{2} → H may be extended to a unitary map of L^{2}(T) onto K by setting

J(\overline{z}^{N}) = E_{−N} for N ≥ 0.

Clearly the subspace K ⊂ L reduces S^{♯}.

Remark 4.9. The orthogonal projection P of L onto H also admits an explicit description. Since J is an isometry of H^{2} onto H, P is the orthogonal projection onto RanJ, so P = JJ^{∗}. Consequently, for {x_{n}} ∈ L′′, P x = Jh where h = J^{∗}x ∈ H^{2} is characterized by

⟨h, g⟩_{H^{2}} = ⟨J^{∗}x, g⟩_{H^{2}} = ⟨x, Jg⟩_{L} = Lim⟨x_{n}, S^{n}g⟩ = Lim⟨S^{∗n}x_{n}, g⟩ ∀g ∈ H^{2},

i.e. h is the ”weak limit with respect to Lim” of the sequence S^{∗n}x_{n}. □

Our next aim is to determine T ∈ B(H^{2}) for which T^{♯}K ⊂ K and T^{♯}↾K = 0, respectively.

Proposition 4.10. Suppose that T ∈ A or that Lim satisfies (C4). Then T^{♯}K ⊂ K if and only if

(31)  ∑_{n∈Z} | Lim_{k} t_{k,n+k} |^{2} = Lim_{k} ∑_{n=0}^{∞}|t_{k,n}|^{2},  where t_{k,n} := ⟨T e_{k}, e_{n}⟩.

Visually, in the matrix of T with respect to the orthonormal basis {e_{n}}, the Lim’s on the left-hand side are Limits of the entries on diagonals parallel to the main diagonal, whereas on the right-hand side is the Limit of norms of the columns of the matrix.

Proof. T^{♯}K ⊂ K is equivalent to T^{♯}E_{N} ∈ K ∀N ∈ N.

For T ∈ A, T^{♯}S^{♯} = S^{♯}T^{♯}, so T^{♯}E_{N} = T^{♯}S^{♯N}E_{0} = S^{♯N}E_{0}; since K reduces S^{♯}, T^{♯}E_{N} ∈ K ⟺ T^{♯}E_{0} ∈ K. Because {E_{N}} is a basis of K, this is in turn equivalent to

(32)  ∥T^{♯}E_{0}∥^{2} = ∑_{M∈N} |⟨T^{♯}E_{0}, E_{M}⟩|^{2}.

Now ∥T^{♯}E_{0}∥^{2} = Lim_{k} ∥T S^{k}e_{0}∥^{2}, which is the right-hand side of (31), while

⟨T^{♯}E_{0}, E_{M}⟩ = Lim_{k}⟨T S^{k}e_{0}, S^{M+k} e_{0}⟩ = Lim_{k}⟨T e_{k}, e_{M+k} ⟩,

and so (32) is equivalent to (31).

For a Lim satifying (C4), we have to check (32) not only for E_{0}, but for all E_{N},; but

∥T^{♯}E_{N}∥^{2} = Lim_{k} ∥T S^{N+k} e_{0}∥^{2} = Lim∥T S^{k}e_{0}∥^{2}

⟨T^{♯}E_{N}, E_{M}⟩ = Lim⟨T S^{N+k} e_{0}, S^{M+k} e_{0}⟩ = Lim⟨T e_{k}, e_{M−N+k} ⟩,

the last equalities on each line being a consequence of (C4); thus we arrive at (31) once again. □

The last criterion is rather discouraging; not only because it is somewhat complicated to check, but also because it depends on the Banach limit chosen. To see this, restrict attention to diagonal operators. If T = diag(c_{n}), c_{n} ∈ C, then (31) reduces to

| Limc_{k}|^{2} = Lim|c_{k}|^{2}.

This always holds if Lim satisfies (C5), i.e. is multiplicative; as we have seen, it cannot satisfy (C4) in that case, and so our criterion says exactly that T^{♯}K ⊂ K for all diagonal operators T belonging to A (i.e. subject to c_{k} − c_{k+1} → 0). On the other hand, if we take c_{k} = (−1)^{k}, then Limc_{k} = 0 for every Lim satisfying (C4), while Lim|c_{k}|^{2} = 1. All one can say for general Lim’s is that | Limc_{k}|^{2} ≤ Lim|c_{k}|^{2}, the equality taking place iff Lim|c_{k} − c|^{2} = 0, where c = Limc_{k}. In fact,

Lim|c_{k} − c|^{2} = Lim (|c_{k}|^{2} − c_{k}\overline{c} − c\overline{c_{k}} + |c|^{2}) =
= Lim|c_{k}|^{2} − Limc_{k} . \overline{c} − c.\overline{Limc_{k}} + |c|^{2} = Lim|c_{k}|^{2} − |c|^{2}.

The criterion for T^{♯}↾K = 0 is much simpler. Define

𝔖̃_{0} := {T ∈ B(H^{2}) : Lim∥T e_{n}∥^{2} = 0}.

If Lim satisfies (C4), T ∈ 𝔖̃_{0} implies that Lim∥T S^{n}f∥^{2} = 0 for all f ∈ H^{2}. This is obvious when f is a polynomial (i.e. a linear combination of e_{n}); for general f ∈ H^{2}, take a polynomial p such that ∥f − p∥ < ϵ and observe that Lim∥T S^{n}(f − p)∥^{2} ≤ ∥T ∥^{2}ϵ^{2}, so Lim∥T S^{n}f∥^{2} ≤ 2∥T ∥^{2}ϵ^{2}; since ϵ > 0 may be arbitrary, the claim follows.

Proposition 4.11. Suppose that T ∈ A or that Lim satisfies (C4). Then T^{♯}↾K = 0 if and only if T ∈ 𝔖̃_{0}.

Proof. T^{♯}↾K = 0 if and only if T^{♯}E_{N} = 0 for all N ∈ Z.
If T ∈ A, T^{♯}E_{N} = S^{♯N}T^{♯}E_{0}, so it suffices to determine when T^{♯}E_{0} = 0. But
∥T^{♯}E_{0}∥^{2} = Lim∥T S^{n}e_{0}∥^{2} = Lim∥T e_{n}∥^{2},

and the assertion follows.

If Lim satisfies (C4), we have

∥T^{♯}E_{N}∥^{2} = Lim_{k} ∥T S^{k+N} e_{0}∥^{2} = Lim∥T S^{k}e_{0}∥^{2},

which vanishes iff T ∈ 𝔖̃_{0}. □

Now let us see how the above apparatus applies to Toeplitz operators on the Hardy space H^{2}. For T = T_{ϕ}, ϕ ∈ L^{∞}(T) ⊂ L^{2}(T), we have ⟨T e_{k}, e_{n}⟩ = ϕ_{n−k} (the Fourier coefficient). Hence, the condition (31) reads

∑_{n∈Z} |ϕ_{n}|^{2} = Lim_{k} ∑_{n=−k}^{∞} |ϕ_{n}|^{2},

which is certainly true; thus, T_{ϕ}^{♯}K ⊂ K. Via the isomorphism J between K and L^{2}(T), the operator T_{ϕ}^{♯}↾K induces some operator on L^{2}(T); let us identify it. One has^{11}

⟨T_{ϕ}^{♯}E_{N}, E_{M}⟩ = Lim_{k}⟨T_{ϕ}S^{k+N} e_{0}, S^{k+M} e_{0}⟩ = Lim_{k} ∫_{T} ϕ(z)z^{k+N} \overline{z}^{k+M} dz = ϕ_{M−N},

and so T_{ϕ}^{♯}J = JM_{ϕ} on L^{2}(T), where M_{ϕ} : L^{2}(T) → L^{2}(T) is the operator of multiplication by ϕ ∈ L^{∞}(T).

Next, since T_{ϕ}^{♯}K ⊂ K for all ϕ ∈ L^{∞}(T), we must also have T^{♯}K ⊂ K for all T ∈ T_{H}, the C^{∗}-algebra generated by τ_{H} = {T_{ϕ} : ϕ ∈ L^{∞}(T)}. The operator T^{♯}↾K then commutes with S^{♯}↾K; passing to L^{2}(T) via the isomorphism J, we see that the operator J^{−1}T^{♯}J commutes with M_{z}, so is of the form M_{ϕ} for some ϕ ∈ L^{∞}(T). Thus, in view of the preceding paragraph, T = T_{ϕ} + X, where X^{♯}↾K = 0, i.e. X ∈ 𝔖̃_{0} (Proposition 4.11). Also, ∥ϕ∥_{∞} = ∥M_{ϕ}∥ = ∥T_{ϕ}^{♯}↾K∥ = ∥T^{♯}↾K∥ ≤ ∥T^{♯}∥ = ∥T ∥_{e} ≤ ∥T ∥, which shows that the linear map ξ : T_{H} → L^{∞}(T) : T ↦ ϕ is multiplicative, because, whenever T_{1}, T_{2} ∈ T_{H},

(T_{1}T_{2})^{♯}↾K = (T_{1}^{♯}↾K)(T_{2}^{♯}↾K) = (JM_{ϕ_{1}}J^{−1})(JM_{ϕ_{2}}J^{−1}) = JM_{ϕ_{1} ϕ_{2}}J^{−1},

and so ξ(T_{1}T_{2}) = ϕ_{1}ϕ_{2} = ξ(T_{1})ξ(T_{2}). Consequently, kerξ must contain the commutator ideal ComT_{H} of the C^{∗}-algebra T_{H}. In fact, a stronger (at first sight) condition holds: kerξ contains not only the commutator [T_{ϕ}, T_{ψ}], but also the semi-commutator [T_{ϕ}, T_{ψ}) = T_{ϕψ} − T_{ϕ}T_{ψ}, for arbitrary ϕ, ψ ∈ L^{∞}(T):

(T_{ϕ}T_{ψ})^{♯}↾K = (T_{ϕ}^{♯}↾K)(T_{ψ}^{♯}↾K) = (JM_{ϕ}J^{−1})(JM_{ψ}J^{−1}) = JM_{ϕψ}J^{−1} = T_{ϕψ}^{♯}↾K,

and so

(33)  ξ(T_{ϕψ} − T_{ϕ}T_{ψ}) = 0.

Remark 4.12. Before going on, let us briefly discuss the relationship between various spaces of operators which have appeared above:

𝔖_{0} = {T ∈ B(H^{2}) : lim_{n→∞} ∥T S^{n}e_{0}∥ = 0} = {T ∈ B(H^{2}) : lim_{n→∞} ∥T S^{n}f∥ = 0 ∀f ∈ H^{2}}

11Here, exceptionally, dz is the arc-length measure on T.

𝔖̃_{0} = {T ∈ B(H^{2}) : Lim∥T S^{n}e_{0}∥^{2} = 0} = (sometimes)
= {T ∈ B(H^{2}) : Lim∥T S^{n}f∥ = 0 ∀f ∈ H^{2}},

A = {T ∈ B(H^{2}) : T − S^{∗}T S ∈ Comp },

τ_{H} = {T_{ϕ} : ϕ ∈ L^{∞}(T)}, T_{H} = C^{∗}-alg(τ_{H}),

ComT_{H} = the commutator ideal of T_{H},

Comp= the compact operators in B(H^{2}).

Consider also

E := {T ∈ B(H^{2}) : ∃ lim_{n→∞} U^{∗n}T S^{n}f, ∀f ∈ H^{2}} = {T ∈ B(H^{2}) : ∃ lim_{n→∞} U^{∗n}T S^{n}e_{0}},

Ẽ:= τ_{H} + 𝔖̃_{0}.

It is well-known that 𝔖_{0} ⊃ ComT_{H} = 𝔖_{0} ∩ T_{H} ⊃ Compand T_{H} = τ_{H} + ComT_{H}; both inclusions are proper. Consequently,

τ_{H} + Comp⊂ τ_{H} + ComT_{H} = T_{H} ⊂ τ_{H} + 𝔖_{0} = E,

both inclusions being proper (the last equality is easily verified). Next, 𝔖_{0} is a proper^{12} subset of 𝔖̃_{0}, and so E is a proper subset of Ẽ. As for A, we have, of course, T_{H} ⊂ A; further, A \ E ≠ ∅ (take the diagonal operator with weights c_{k} = exp(i ln(k + 1))) and 𝔖̃_{0} \ A ≠ ∅ (find a sequence {c_{k}}_{k∈N} , consisting only of infinitely many zeroes and infinitely many 1’s, and such that Limc_{k} = 0; then take diag(c_{n})). The author does not know if the sets E \ A, Ẽ \ A, A \ Ẽ and 𝔖_{0} \ A are empty or not. As for the Toeplitz operators on the Bergman space A^{2}(D) (remember we have agreed to denote them B_{ϕ} throughout this chapter), the spaces

τ_{B} = {B_{ϕ} : ϕ ∈ L^{∞}(D)}, T_{B} = C^{∗}-alg (τ_{B}),

satisfy Comp⊂ T_{B} ⊂ A (the first inclusion is proper, and so is probably the second) and τ_{B} \ E ≠ ∅ (the operator which has served as a counterexample to A ⊂ E equals in fact B_{ϕ} for a suitable ϕ ∈ L^{∞}(D), cf. Example 4.2). These assertions remain in force for the Fock space A^{2}(C) in place of A^{2}(D) (cf. also Example 4.2). □

In the last-but-one paragraph, we have almost recovered the decomposition of T_{H} and the symbol map which were mentioned at the beginning of this chapter. Let us try to carry out the same procedure for the algebra A.

Theorem 4.13. Let T ∈ A and suppose T^{♯}K ⊂ K. Then T admits a unique decomposition of the form T = T_{ϕ} + R, where ϕ ∈ L^{∞}(T), ∥ϕ∥_{∞} ≤ ∥T ∥_{e} ≤ ∥T ∥, and R ∈ 𝔖̃_{0}. Besides, the map T ↦ ϕ is multiplicative, i.e. if V ∈ A, V^{♯}K ⊂ K, V = T_{ψ} + Q, then T V ↦ ϕψ. Consequently, T V − V T ∈ 𝔖̃_{0}.

Proof. Repeat verbatim what was said for the case of T_{H}: T^{♯}K ⊂ K ⟹ J^{−1}T^{♯}J : L^{2}(T) → L^{2}(T), T ∈ A ⟹ J^{−1}T^{♯}J commutes with M_{z}, so is of the form M_{ϕ}, ϕ ∈ L^{∞}(T). The operator R := T − T_{ϕ} then satisfies R^{♯}↾K = 0, i.e. R ∈ 𝔖̃_{0}. ∥ϕ∥_{∞} = ∥M_{ϕ}∥ = ∥T^{♯}↾K∥ ≤ ∥T^{♯}∥ = ∥T ∥_{e} ≤ ∥T ∥, whence also the uniqueness assertion follows. Finally, (T V)^{♯}↾K = (T^{♯}↾K)(V^{♯}↾K) = (JM_{ϕ}J^{−1})(JM_{ψ}J^{−1}) = JM_{ϕψ}J^{−1} = (V T)^{♯}↾K. □

Thus, things go well when T^{♯}K ⊂ K, and this happens iff T ∈ τ_{H} + 𝔖̃_{0} = Ẽ (and T ∈ A). In general, we have only the following weaker result.

Proposition 4.14. Let T ∈ A. Then T = T_{ϕ} + R, where ∥ϕ∥_{∞} ≤ ∥T ∥_{e} ≤ ∥T ∥ and P (K)R^{♯}↾K = 0.

The decomposition is unique.

Proof. T ∈ A ⟹ T^{♯}S^{♯} = S^{♯}T^{♯}; denote W = P (K)T ↾K. Then W K ⊂ K and W S^{♯}↾K = P (K)T S^{♯}↾K = S^{♯}W, because S^{♯}P (K) = P (K)S^{♯} (since K reduces S^{♯}). This implies, once again, J^{−1}W J =

12To see that the inclusion is proper, take a diagonal operator with weights c_{k} such that Limc_{k} = 0 and limc_{k} does not exist.

M_{ϕ}, ϕ ∈ L^{∞}(T), and the operator R := T − T_{ϕ} must satisfy P (K)R^{♯}↾K = 0. The norm estimate is proved in the same way as before. □

Note that, however, this map T ↦ ϕ is no longer multiplicative. The operators R such that P (K)R^{♯}↾K = 0 admit an easy characterization: their matrices have entries which ”tend to zero” along every diagonal parallel to the main diagonal.

Proposition 4.15. Assume that R ∈ A or that Lim satisfies (C4). Then P (K)R^{♯}↾K = 0 if and only if

(34)  Lim_{k}⟨Re_{k}, e_{N+k} ⟩ = 0  ∀N ∈ Z.

Proof. P (K)R^{♯}↾K = 0 iff ⟨R^{♯}E_{N}, E_{M}⟩ = 0 ∀M, N ∈ Z. If R ∈ A, this is equivalent to ⟨R^{♯}E_{0}, E_{M}⟩ = 0 ∀M ∈ Z, which is exactly (34). In the other case,

⟨R^{♯}E_{N}, E_{M}⟩ = Lim_{k}⟨Re_{N+k} , e_{M+k} ⟩ = Lim_{k}⟨Re_{k}, e_{M−N+k} ⟩,

the last equality being a consequence of (C4). □

We close this chapter with two observations. The first concerns the Allan-Douglas localization principle [12]. Because T^{♯}S^{♯} = S^{♯}T^{♯} if T ∈ A, the space ker(S^{♯} − μ) = ker(S^{♯∗} − \overline{μ}) is invariant under T^{♯} when T ∈ A, for every μ ∈ T. Denote I_{μ} the (closed, two-sided, *-) ideal in A generated by Compand S − μ. Let A_{μ} be the C^{∗}-algebra A/I_{μ}. Similar construction may be applied to the algebras T_{B} and T_{F}.

Theorem 4.16. The mapping W ↦ (W + I_{μ})_{μ∈T} from A/Compinto the C^{∗}-direct sum ⨁_{μ∈T} A_{μ} is an

(isometrical) C^{∗}-isomorphism of A/Componto a C^{∗}-subalgebra of ⨁_{μ∈T} A_{μ}.

The same assertion is valid for T_{B} and T_{F} in place of A.

Proof. Follows from the general Allan-Douglas theory; for instance, apply [7, proof of Theorem 1.34–d.] with A = A/Comp , B = {f(s) : f ∈ C(T)}, where s = π(S) is the image of S in Calk(H^{2}), N = {f(s) : f ∈ C(T), f(μ) = 0}, J_{N} = I_{μ}/Comp , A_{N} = A_{μ}/Comp . □

Since, as was pointed out at the beginning of this chapter, Comp⊂ ComA, multiplicative linear functionals on A correspond bijectively to those on A/Comp . If ϕ is one of the latter, then ϕ(s) ∈ σ(s) = σ_{e}(S) = T, and so ϕ(S − μ) = 0 for some μ ∈ T, i.e. kerϕ ⊃ I_{μ}/Comp , so ϕ induces a multiplicative linear functional on A_{μ}/Compand hence on A_{μ}. On the other hand, every multiplicative linear functional on A_{μ} yields a multiplicative linear functional on A. Thus, the problem of existence of multiplicative linear functionals on A can be reduced to the same problem for A_{μ}. Unfortunately, the latter seems to be equally hopeless to solve. We conclude this small digression with a proposition which shows that it suffices to deal with A_{1}.

Lemma 4.17. For ϵ ∈ T and f a function on D or C, let (R_{ϵ}f)(z) := f(ϵz). Then R_{ϵ}^{−1} = R\overline{ϵ}, R_{ϵ} is

a unitary (i.e. isometric and onto) operator on A^{2}(D), L^{∞}(D), A^{2}(C) and L^{∞}(C), and the transformation T ↦ R_{ϵ}^{∗}T R_{ϵ} is an isometry on both B(A^{2}(D)) and B(A^{2}(C)) which maps Compinto Comp , A onto A, and B_{ϕ} into B_{R\overline{ϵ}ϕ}, resp. F_{ϕ} into F_{R\overline{ϵ}ϕ}.

Proof. Straightforward. □

Proposition 4.18. For any μ, ν ∈ T, A_{μ} ≃ A_{ν} (isometrical C^{∗}-isomorphism).

Proof. The transformation T ↦ R_{ϵ}^{∗}T R_{ϵ} maps A onto A, Componto Comp , and S − μ into \overline{ϵ}S − μ; consequently, it must map I_{μ} onto I_{ϵμ} and A_{μ} onto A_{ϵμ}. The map is, moreover, clearly an isometrical C^{∗}-isomorphism. □

The second observation applies only to the one-dimensional Fock space A^{2}(C), so imagine H^{2} to be identified with A^{2}(C) by means of the unitary operator W_{F} , as was remarked at the beginning of this chapter. Let T̃, once again, denote the Berezin transform of the operator T ∈ B(A^{2}(C)), i.e.

T̃(λ) = ⟨T k_{λ}, k_{λ}⟩, k_{λ}(z) = exp(\frac{\overline{λ}z}{2} − \frac{|λ|^{2}}{4} ) ∈ A^{2}(C),

and define T̂ := F_{T}, the Toeplitz operator on A^{2}(C) with symbol T̃. Owing to the fact that, unlike D, the group of isometries of the Euclidean metric on C admits a commutative transitive subgroup, Berger and Coburn [6] have shown that

T̂ = ∫_{C} W_{α}^{∗}T W_{α} dμ(α),

where the integral is the weak (Gelfand-Pettis) integral, and W_{α} are unitary operators on A^{2}(C) given by

(W_{α}f)(z) = k_{α}(z)f(z − α), α ∈ C.

They proved the following lemma (attributed to W.Zame).

Lemma 4.19. Let H be a separable Hilbert space, (X, ν) a finite measure space, and F : X → B(H) a norm-bounded, WOT-measurable function (i.e. x ↦ ⟨F (x)f, g⟩ is ν-measurable ∀f, g ∈ H), whose values are compact operators. Then the weak (Gelfand-Pettis) integral

∫_{X} F (x) dν(x)

(exists and) is also a compact operator.

Proof. See [6], Lemma 12; we reproduce the proof here for convenience. First, for arbitrary f, g ∈ H, the function ⟨F (x)f, g⟩ is bounded and ν-measurable, so the integral ∫_{X}⟨F (x)f, g⟩ dν(x) exists and its modulus does not exceed

ν(X) . sup_{X} ∥F (x)∥ . ∥f∥.∥g∥.

Consequently, the weak integral

∫_{X} F (x) dν(x)

exists and its norm is bounded by ν(x) . sup_{X} ∥F (x)∥.

Pick a basis {e_{j}}_{j∈N} of H and let P_{k} stand for the projection onto the span of e_{0}, e_{1}, …, e_{k}. Denote

E_{k} = {x ∈ X : ∥P_{k}F (x) − F (x)∥ < ϵ}.

Note that ⋃_{k∈N} E_{k} = X since F takes values in Comp . Further, the function x ↦ ∥P_{k}F (x) − F (x)∥ is

measurable, since it equals

(35)  sup|⟨(P_{k}F (x) − F (x))f, g⟩|,

the supremum being taken over f, g in a countable dense subset of the unit ball of H. Consequently, the sets

E′_{k} = E_{k} \ ⋃_{j=0}^{k−1} E_{j}

are measurable, disjoint, and their union is all of X. Choose m so large that

∑_{k>m} ν(E′_{k}) < ϵ.

Then

∫_{X} F (x) dν(x) = ∑_{k=1}^{m}P_{k} ∫_{E′_{k}} F (x) dν(x)+

∑_{k=1}^{m}∫_{E′_{k}}[F (x) − P_{k}F (x)] dν(x) + ∫_{⋃_{k>m} E′_{k}} F (x) dν(x).

Since P_{k} ∈ Comp , the first summand belongs to Comp , while the norms of the second and the third do not exceed ϵν(X) and ϵ sup_{x∈X} ∥F (x)∥, respectively. As ϵ > 0 was arbitrary, the assertion of the lemma follows. □

Corollary 4.20. If K ∈ Comp , then K̂ ∈ Comp .

Proof. Apply the lemma to X = C, ν = μ (the Gaussian measure) and F (x) = W_{α}^{∗}KW_{α}. The function F (α) is even WOT-continuous, since α ↦ W_{α} is SOT continuous. To prove the latter, it suffices to check that α ↦ W_{α}g_{λ} is continuous for each λ ∈ C (because the linear combinations of g_{λ} are dense in A^{2}(C), cf. Lemma 2.5), which is easily verified. □

Corollary 4.21. If T ∈ A, then T̂ ∈ A.

Proof. We have

T̂ − S^{∗}T̂S = ∫_{C}(W_{α}^{∗}T W_{α} − S^{∗}W_{α}^{∗}T W_{α}S) dμ(α).

However,

(36)  W_{α} = B_{exp ( \frac{|α|^{2}}{4} +iIm(\overline{α}z))} ,  (see [6], section 3),

which implies W_{α}^{∗}T W_{α} ∈ A whenever T ∈ A. It remains to apply the lemma to X = C, ν = μ and F (α) = W_{α}^{∗}T W_{α} − S^{∗}W_{α}^{∗}T W_{α}S; again, F is even WOT-continuous. □

The last two corollaries prompt to formulate the following

Conjecture. (a) If T ∈ ComA, then T̂ ∈ ComA.

(b) For all T ∈ A, the operator T − T̂ belongs to ComA. Both assertions could be proved if we could replace B(H) and Compin Lemma 4.19 by A and ComA,

respectively. (Indeed, according to the definition of T̂,

T − T̂ = ∫_{C} W_{α}^{∗}[W_{α}, T] dμ(α),

and the integrand is bounded by 2∥T ∥, WOT-continuous, and, in view of (36), belongs to ComA for every α ∈ C.) The author suspects that the lemma remains in force even if B(H) and Compare replaced by an arbitrary C^{∗}-algebra 𝔄⊂ B(H) and a closed, two-sided *-ideal I in 𝔄. The above approach of W.Zame works whenever the C^{∗}-algebra I has a countable approximate identity, or if X is a separable topological space, ν a Borel measure, and the function x ↦ ∥QF (x)∥ is continuous for every operator Q ∈𝔄, 0 ≤ Q ≤ I. Regarding the conjecture above, the former possibility (a countable approximate identity for ComA) is highly improbable, while the latter seems quite likely indeed to take place, although I have not been able to prove it (i.e. to prove that the function

α ↦ ∥QW_{α}^{∗}T W_{α}∥, α ∈ C,

is continuous; it is easy to see from (35) that it is lower semicontinuous, which, however, is not enough for our purposes).

Thus, in the abelianized algebra A_{ab} := A/ComA, the transformation T ↦ T̂ would act as the identity; consequently, some insight into the structure of that algebra (multiplicative linear functionals etc.) could be

gained by studying the iterated transforms T ↦ T̂ ↦ ˆ T̂ ↦ ˆ ˆ T̂ ↦ … or, which amounts to the same, of the iterated Berezin transforms f ↦ f̃ ↦ f̃^{(2)} ↦ f̃^{(3)} ↦ … on L^{∞}(C). The latter is equivalent to the study of the asymptotics of the heat equation, since

f̃^{(k)}(z) = u(k/2, z),

where u(t, z) is the solution on (0, +∞) × C of the heat equation \frac{∂u}{∂t} = 4Δ u(t, z) with initial condition u(0, z) = f(z). (See [6] again, or compute directly.) This is one of motivations for undertaking a further study of the Berezin transform, which is the objective of the subsequent chapter.

Chapter 5. THE BEREZIN TRANSFORM

Since the Berezin transform seemingly plays an important role in the theory of Toeplitz operators on the Bergman space and the Fock space, it will be studied in a greater detail in this chapter. Three main topics will be discussed. First, we investigate the boundedness of the Berezin transform on various L^{p} spaces. The connection of the transform with the Laplace and the Laplace-Beltrami operator, respectively, is established. Second, we prove that the fixed points of the Berezin transform are exactly the harmonic functions. Third, some ergodicity properties are discussed; the motivation for this comes from the considerations at the end of Chapter 4. The results are stated for A^{2}(D) and A^{2}(C) only; generalizations to other domains are likely to be possible, but seem to be closer in spirit to Riemannian geometry rather than functional analysis. We begin by recalling the definitions and basic properties.

If f ∈ L^{1}(D, dν), the Berezin transform of f is, by definition,

f̃(w) = ⟨fk_{w}, k_{w}⟩ = ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} f(z) dν(z), w ∈ D,

where k_{w} is the normalized reproducing kernel at w ∈ D:

k_{w}(z) = \frac{g_{w}(z)}{∥g_{w}∥} = \frac{1 − |w|^{2}}{(1 − \overline{w}z)^{2}} .

Note that k_{w} ∈ L^{∞}(D) ∀w ∈ D, so the definition makes sense.

Similarly, if f ∈ L^{1}(C^{N}, dμ), define f̃ to be

f̃(w) = ⟨fk_{w}, k_{w}⟩ = ∫_{C^{N}} e^{−|w−z|^{2} /2}f(z) \frac{dz}{(2π)^{N}} , w ∈ C^{N},

where^{13}

k_{w} = \frac{g_{w}(z)}{∥g_{w}∥} = e^{\frac{\overline{w}z}{2}− \frac{|w|^{2}}{4}} .

(Since it will always be clear whether we are discussing A^{2}(C^{N}) or A^{2}(D), no ambigiuty concerning g_{w} and k_{w} should arise.)

For typographical reasons, the Berezin transform f̃ is sometimes also denoted Bf.

Proposition 5.1. f̃ is an infinitely differentiable function on D, resp. C^{N}.

Proof. Differentiate under the integral sign. (See the proof of Proposition 5.20 for details.) □

Proposition 5.2. If f is bounded, then so is Bf ≡ f̃ and ∥f̃∥_{∞} ≤ ∥f∥_{∞}. In other words, B is a contraction in L^{∞}(D), resp. L^{∞}(C^{N}).

Proof. |f̃(w)| ≤ ∥fk_{w}∥_{2}∥k_{w}∥_{2} ≤ ∥f∥_{∞}.∥k_{w}∥_{2}^{2} = ∥f∥_{∞}. □

Remark 5.3. Since f = f̃ when f is a constant function, the norm of B on L^{∞}(C^{N}) or L^{∞}(D) is, in fact, equal to one. □

Proposition 5.4. If f ≥ 0, then f̃ ≥ 0; if f ≥ g, then f̃ ≥ g̃.

Proof. B is an integral operator with positive kernel. □

We are going to show that the Berezin transform is also a contractive linear operator on other L^{p} spaces, provided they are taken with respect to an appropriate measure — namely, the measure which is intrinsic for the Riemannian geometry of the domain. Recall that the Lebesgue measure dz on C^{N} is (up

13Here, as usual, \overline{w}z = ∑_{j=1}^{N} \overline{w}_{i}z_{i}, |w|^{2} = \overline{w}w, etc. to multiplication by a constant factor) the only measure invariant with respect to the group of the rigid motions of C^{N}. Similarly, on D, the only measure left invariant by all Möbius transformations

(37)  z ↦ ϵ\frac{z − w}{1 − \overline{w}z} := ϵω_{w}(z) (w ∈ D, ϵ ∈ T)

is the pseudo-hyperbolic measure

dη(z) := \frac{dν(z)}{(1 − |z|^{2})^{2}} .

(The invariance may be verified by direct computation.) It turns out that the Berezin transform behaves well with respect to the invariant measures.

Proposition 5.5. (a) The mapping B : f ↦ f̃ is a contractive linear operator on each of the spaces

L^{p}(D, dη(z)), 1 ≤ p ≤ ∞.

(b) Similar assertion holds for L^{p}(C^{N}, dz), 1 ≤ p ≤ ∞.

Proof. (a) Since L^{1}(D, dη) ⊂ L^{1}(D, dν), the Berezin transform is defined on the former space, and

|f̃(w)| = |∫_{D} f(z)\frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(z)| ≤ B(|f|)(w).

Thus

∫_{D} |f̃(w)| \frac{dν(w)}{(1 − |w|^{2})^{2}} ≤∫_{D} (∫_{D} |f(z)| \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(z)) \frac{dν(w)}{(1 − |w|^{2})^{2}} =
= ∫_{D} |f(z)| ∫_{D} \frac{dν(w)}{|1 − \overline{w}z|^{4}} dν(z) =
= ∫_{D} |f(z)| . ⟨g_{w}, g_{w}⟩ dν(z) = ∫_{D} |f(z)| \frac{dν(z)}{(1 − |z|^{2})^{2}} ,

the change of the order of integration being justified by the positivity of the integrand. It follows that B is a contraction on L^{1}(D, dη). The same is true for L^{∞} (Proposition 5.2), and so the result follows from the Marcinkiewicz interpolation theorem. The proof of (b) is similar. □

The last proposition suggests that there might be some closer relationship between the Berezin transform and the Riemannian geometry on D, resp. C^{N}. This is indeed the case. Before clarifying this point, we establish an alternative formula for f̃.

Proposition 5.6. (a) In the notation of (37), for arbitrary f ∈ L^{1}(D, dν),

f̃(w) = ∫_{D} f(ω_{−w}(y)) dy.

(b) Similarly,

f̃(w) = ∫_{C^{N}} f(y + w) dμ(y)

for every f ∈ L^{1}(C^{N}, dμ).

Proof. If we make an (analytic) change of coordinates

y = ω_{w}(z), i.e. z = ω_{−w}(y),

then

dν(y) = |ω′_{w}|^{2} . dν(z) = \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(z),

and so

f̃(w) = ∫_{D} f(z) \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(z) = ∫_{D} f(ω_{−w}(y)) dν(y)

as claimed. (b) is similar, only even simpler. □

Proposition 5.7. The Berezin transform commutes with the ”group of rigid motions” of D, resp. C^{N}. More precisely,

B(f ∘ ω_{a}) = (Bf) ∘ ω_{a}

for every f ∈ L^{1}(D, dν), a ∈ D, while

B(f ∘ t_{a}) = (Bf) ∘ t_{a}

for every f ∈ L^{1}(C^{N}, dμ), a ∈ C^{N}; here t_{a}(z) := z − a.

Proof. For f ∈ L^{1}(D, dν) and a, w ∈ D,
B(f ∘ ω_{a})(w) = ∫_{D} f(ω_{a}(ω_{−w}(z))) dν(z)
= ∫_{D} f(ω_{a}(ω_{−w}(\frac{1 + \overline{a}w}{1 + a\overline{w}} z))) dν(z)
= ∫_{D} f(ω_{\{_{\frac{1+a\overline{w}}{1+\overline{a}w}} ω_{w}(a)\}}(z)) dν(z)

= f̃( − \frac{1 + a\overline{w}}{1 + \overline{a}w} ω_{w}(a))
= f̃(−ω_{−a}(−w)) = f̃(ω_{a}(w)),

since

ω_{a}(ω_{−w}(ϵz)) = ϵ\frac{1 + a\overline{w}}{1 + \overline{a}w} . ω_{\{\overline{ϵ}ω_{w}(a)\}}(z) for ϵ ∈ T, a, w ∈ D

and

\frac{1 + a\overline{w}}{1 + \overline{a}w} ω_{−w}(a) = ω_{−a}(w) = −ω_{a}(−w).

The proof for C^{N} is, once again, similar but much simpler, because the translations t_{a} on C^{N}, unlike Möbius transformations, commute. □

The last proposition has important consequences, since operators commuting with the (Möbius or Euclidean) translations may be described explicitly. Consider the Laplace operator

Δ := ∏_{j=1}^{N} 4\frac{∂^{2}}{∂z_{j}∂\overline{z}_{j}}

on C^{N}, and the Laplace-Beltrami operator^{14}

Δ_{h} := (1 − |z|^{2})^{2} \frac{∂^{2}}{∂z∂\overline{z}}

on D. These operators are symmetric on the subspace of L^{2}(C^{N}, dz) or L^{2}(D, dη), respectively, consisting of infinitely differentiable functions with compact support; since their coefficients are real, they can be extended to (unbounded) selfadjoint operators on the respective L^{2} spaces. Besides, direct calculation reveals that they commute with the group of motions of C^{N} and D, respectively:

Δ(f ∘ t_{a}) = (Δf) ∘ t_{a}, Δ_{h}(f ∘ ω_{a}) = (Δ_{h}f) ∘ ω_{a}.

14Sometimes Δ_{h} is defined as four times this operator.

According to a fundamental work of Gelfand [17], a sort of converse also holds : every operator on L^{2}(C^{N}, dz) or L^{2}(D, dη), commuting with the group of motions, must be a function of Δ or Δ_{h}, respectively. In view of Proposition 5.7, this applies, in particular, to the Berezin transform — it must be a function of Δ, resp. Δ_{h}. This idea goes back to Berezin [5], who even exhibited an explicit formula for B (on D) in terms of Δ_{h}: f̃ = F (Δ_{h})f where

(38)  F (x) = ∏_{n=1}^{∞} (1 − \frac{x}{n(n + 1)} )^{−1} = \frac{πx}{sinπ (\sqrt{x + \frac{1}{4}} − \frac{1}{2})}.

In the case of C^{N}, an explicit formula has been established by Berger and Coburn [6]:

(39)  f̃ = e^{Δ/2} f,

i.e. f̃ is the solution of the heat equation with the initial condition f at the time 1/2. These formulas, although established here only for f ∈ L^{2}(C^{N}, dz) or f ∈ L^{2}(D, dη), respectively, can be shown to (be meaningful and) hold actually for wider classes of functions — this can be done e.g. by appealing to the theory of pseudodifferential operators; we won’t, however, pursue this matter further.

The spaces L^{p}(C^{N}, dz) and L^{p}(D, dη) are rather small — they don’t even contain (nonzero) constant functions. A question which comes into mind is whether the Berezin transform is not actually a bounded linear operator on the spaces L^{p} with respect to the other natural measure — namely, L^{p}(C^{N}, dμ) or L^{p}(D, dν), respectively. This turns out to be true whenever p > 1. Before presenting the proof, we are going to show how the machinery of interpolation spaces may be used to obtain a weaker result. We shall temporarily restrict our attention to D, since most proofs work, with minor modifications, for C^{N} as well.

To prove that B is a bounded operator on L^{p}(D, dν), 1 < p < ∞, it would suffice to prove this fact for p = 1 — since B is a contraction on L^{∞}(D), we could apply the Marcinkiewicz interpolation theorem. Unfortunately, this approach will not work.

Proposition 5.8. B is not a bounded operator on L^{1}(D, dν).

Proof. If it were, its adjoint B^{d} ≡ C,

(40)  (Cf)(z) = ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} f(w) dν(w),  z ∈ D,

would be a bounded operator on L^{∞}(D). It will be shown below (in course of the proof of Lemma 5.13(b) ) that

\frac{1}{2π} ∫_{0}^{2π} \frac{1}{|1 − \overline{z}re^{it}|^{4}} dt = \frac{1 + |z|^{2}r^{2}}{(1 − |z|^{2}r^{2})^{3}} = ∑_{n=0}^{∞}(n + 1)^{2}r^{2n}|z|^{2n},

for z ∈ D and r ∈ (0, 1). Consequently,

(C1)(z) = ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(w) = ∫_{0}^{1}(1 − r^{2})^{2} \frac{1}{2π} ∫_{0}^{2π} |1 − \overline{z}re^{it}|^{−4} dt 2r dr
= ∫_{0}^{1}(1 − r^{2})^{2} ∑_{n=0}^{∞}(n + 1)^{2}r^{2n}|z|^{2n} 2r dr

= ∫_{0}^{1} ∑_{n=0}^{∞}(n + 1)^{2}(1 − t)^{2}t^{n}|z|^{2n} dt

= ∑_{n=0}^{∞} \frac{2(n + 1)}{(n + 2)(n + 3)} |z|^{2n}.

As |z| ↗ 1, this expression behaves (asymptotically) like − log(1 − |z|^{2}); hence C1 ∉ L^{∞}(D), so C ≡ B^{d} cannot be a bounded operator on L^{∞}(D). □

All the same, the above method may be exploited to prove that B is a bounded operator from L^{p}(D, dν) into L^{q}(D, dν) whenever q < p.

Lemma 5.9. The integral operator C given by (40) is a contraction on L^{1}(D, dν) which maps L^{∞}(D) boundedly into L^{p}(D, dν) for every p ∈ ⟨1, ∞). Similar assertion is valid for C^{N}.

Proof. For arbitrary f ∈ L^{1}(D, dν),
∫_{D} |Cf(z)| dν(z) ≤∫_{D} ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} |f(w)| dν(w) dν(z)
= ∫_{D} |f(w)| ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(z) dν(w)  (by Fubini)
= ∫_{D} |f(w)| . ⟨k_{w}, k_{w}⟩ dν(w) = ∫_{D} |f(w)| dν(w),

so C is a contraction on L^{1}(D, dν). If f ∈ L^{∞}(D), then

|Cf(z)| ≤ ∥f∥_{∞} . ∫_{D} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} dν(w) = ∥f∥_{∞} . |C1(z)|.

Hence, to prove the second assertion of the lemma, it suffices to check that C1 belongs to L^{p}(D, dν) for each p ∈ ⟨1, ∞). In course of the proof of the preceding proposition, we have observed that C1(z) behaves like − log(1 − |z|^{2}) as |z| ↗ 1, so it is enough to show that log(1 − |z|^{2}) ∈ L^{p}(D, dν) ∀p ∈ ⟨1, ∞). But

∫_{D} | log(1 − |z|^{2})|^{p} dν(z) = ∫_{0}^{1} | log(1 − r^{2})|^{p} 2r dr = ∫_{0}^{1} | log(1 − t)|^{p} dt = ∫_{0}^{1} | logt|^{p} dt,

and, changing the variable to y = − logt, this reduces to

∫_{0}^{+∞} y^{p}e^{−y} dy = Γ(p + 1) < +∞.

The proof for C^{N} is similar. □

Theorem 5.10. (a) If 1 ≤ q < p ≤ ∞, then B is a bounded operator from L^{p}(D, dν) into L^{q}(D, dν).

(b) The same assertion holds for L^{p} and L^{q} of (C^{N}, dμ).

Proof. We shall deal only with D, the other case being similar. Consider the integral operator C given by (40). By the previous lemma, C is a bounded operator from L^{1}(D, dν) into L^{1}(D, dν) and from L^{∞}(D) into L^{p}(D, dν), ∀p ∈ ⟨1, ∞). According to the Marcinkiewicz interpolation theorem, it must be a bounded operator from L^{q}(D, dν) into L^{r}(D, dν) ∀r ∈ ⟨1, q) for arbitrary q ∈ ⟨1, ∞⟩. It follows that its adjoint, which is exactly B, is a bounded map from L^{r′}(D, dν) into L^{q′}(D, dν) whenever q′ ∈ (1, ∞⟩ and r′ ∈ (q′, ∞⟩; as L^{q′}(D, dν) is boundedly imbedded in L^{1}(D, dν) for arbitrary q′ ≥ 1, we may take even q′ ∈ ⟨1, ∞⟩. Changing slightly the notation produces the assertion of the theorem. □

Remark 5.11. We have proved actually a little more. Recall that the space X_{1}(D) is, by definition, the class of all functions f on D such that f ∈ L^{p}(D, dν) for all 1 ≤ p < ∞ and

∥f∥_{X} := sup_{1≤p<∞} \frac{∥f∥_{p}}{p} < +∞.

Equipped with the norm ∥ . ∥_{X}, X_{1}(D) becomes a Banach space (cf. [21], section 4.8). In course of the proof of Lemma 5.9, we have almost proved that C maps L^{∞}(D) into X_{1}(D). Indeed, it suffices, as above, to verify that C1 ∈ X_{1}(D); this is reduced to the assertion that log(1 − |z|^{2}) ∈ X_{1}(D), and this in turn to logt ∈ X_{1}(0, 1), i.e. to the assertion that

(41)  sup_{p≥1} \frac{Γ(p + 1)^{1/p}}{p} < +∞.

But, owing to Stirling’s formula,

Γ(p + 1)^{1/p} ∼ \frac{p}{e} . (2πp)^{\frac{1}{2p}} ,

so (41) is true. Thus, C is a bounded map from L^{∞}(D) into X_{1}(D).

The Marcinkiewicz interpolation theorem asserts that if a linear operator T maps (boundedly) L^{∞}(D) → L^{∞}(D) and L^{1}(D, dν) → L^{1}(D, dν), it must map L^{p}(D, dν) boundedly into L^{p} ∀p ∈ (1, ∞). The first condition may be relaxed^{15} to L^{∞}(D) → BMO(D); the space BMO(D) is bigger than L^{∞}(D), but still lies in all L^{q}(D, dν), 1 ≤ q < ∞. The space X_{1} is, in turn, bigger than BMO while still lying in all L^{q}, 1 ≤ q < ∞. If we knew that the condition L^{∞} → BMO may be further relaxed to L^{∞} → X_{1}, we could conclude that C would be a bounded map from L^{p} into L^{p} ∀p ∈ (1, ∞) — and, consequently, so would be the Berezin transform B, the adjoint of C. Whether the condition may indeed be relaxed like this seems to be an interesting unsolved problem from interpolation theory. □

To prove that the Berezin transform B is actually bounded on L^{p}(D, dν), 1 < p < ∞, we employ a generalization of the classical Schur test.

Proposition 5.12. Let (X, dx) and (Y, dy) be measure spaces, k(x, y) a nonnegative measurable function on X × Y, 1 < p < ∞, q = \frac{p}{p − 1}. If P and Q are positive measurable functions on X and Y,

respectively, and α, β positive numbers such that

∫_{Y} k(x, y)Q(y)^{q/p} dy ≤ αP (x) dx-almost everywhere on X,

∫_{X} k(x, y)P (x)^{p/q} dx ≤ βQ(y) dy-almost everywhere on Y,

then the integral operator T : L^{p}(Y, dy) → L^{p}(X, dx),

(T g)(x) := ∫_{Y} k(x, y)g(y) dy,

is bounded and ∥T ∥_{p→p} ≤ α^{1/q}β^{1/p}.

Proof. Let g ∈ L^{p}(Y, dy). Then

∫_{X} (∫_{Y} k(x, y)|g(y)| dy)^{p} dx = ∫_{X} (∫_{Y} [\frac{k(x, y)}{Q(y)}|g(y)|^{p}]^{1/p} [k(x, y)^{1/q}Q(y)^{1/q}] dy) dx

(by Hölder’s inequality)  ≤ ∫_{X} (∫_{Y} \frac{k(x, y)}{Q(y)}|g(y)|^{p} dy . [∫_{Y} k(x, y)Q(y)^{q/p} dy]^{p/q}) dx

≤ ∫_{X} (∫_{Y} \frac{k(x, y)}{Q(y)}|g(y)|^{p} dy . α^{p/q}P (x)^{p/q}) dx
(by Fubini)  = ∫_{Y} ∫_{X} \frac{α^{p/q}}{Q(y)}|g(y)|^{p} . P (x)^{p/q}k(x, y) dx dy
≤ ∫_{Y} \frac{α^{p/q}}{Q(y)}|g(y)|^{p} . βQ(y) dy = α^{p/q}β . ∥g∥_{p}^{p}.

It follows that ∥T g∥_{p} ≤ ∥T (|g|)∥_{p} ≤ α^{1/q}β^{1/p}∥g∥_{p}, as asserted. □

Lemma 5.13. (a) If a ∈ (−1, 0), then

∫_{0}^{1} \frac{(1 − t)^{a}}{1 − Rt} dt ≤ \frac{1}{(−a)(a + 1)} . (1 − R)^{a} ∀R ∈ ⟨0, 1).

15cf. [21], paragraph 4.10.6, and the references given therein.

(b) If a ∈ (−1, 2), then

∫_{D} \frac{(1 − |x|^{2})^{a}}{|1 − x\overline{y}|^{4}} dν(x) ≤ C_{a} . (1 − |y|^{2})^{a−2} ∀y ∈ D,

where C_{a} is a (finite) number depending only on a. Proof. (a) Making several changes of the variable yields

∫_{0}^{1} \frac{(1 − t)^{a}}{1 − Rt} dt = ∫_{0}^{1} \frac{w^{a}}{(1 − R) + Rw} dw  (w = 1 − t, dt = −dw)
= − \frac{1}{a} ∫_{1}^{+∞} \frac{s^{1/a} ds}{(1 − R) + Rs^{1/a}}  (s = w^{a}, dw = \frac{1}{a}s^{\frac{1}{a} −1} ds)
= c ∫_{1}^{+∞} \frac{ds}{R + (1 − R)s^{c}}  (u = (1 − R)^{1/c}s, ds = (1 − R)^{−1/c} du)
= c(1 − R)^{−1/c} ∫_{(1−R)^{1/c}}^{+∞} \frac{du}{R + u^{c}} ,

where c = −\frac{1}{a} ∈ (1, +∞); but

∫_{1}^{+∞} \frac{du}{R + u^{c}} ≤ ∫_{1}^{+∞} \frac{du}{u^{c}} = \frac{1}{c − 1},

∫_{(1−R)^{1/c}} \frac{du}{R + u^{c}} ≤ ∫_{(1−R)^{1/c}}^{1} du ≤ 1,

so, indeed,

∫_{0}^{1} \frac{(1 − t)^{a}}{1 − Rt} dt ≤ c(1 − R)^{−1/c} . \frac{c}{c − 1} = \frac{1}{(−a)(a + 1)} (1 − R)^{a}.

(b) Let r ∈ (0, 1) and denote, for brevity, ρ = |\overline{y}r| and R = |y|^{2}. By virtue of the Residue theorem,

\frac{1}{2π} ∫_{0}^{2π} \frac{1}{|1 − \overline{y}re^{it}|^{4}} dt = \frac{1}{2π} ∫_{0}^{2π} |1 − ρe^{it}|^{−4} dt =

= \frac{1}{2π} ∫_{0}^{2π}(1 − ρe^{it})^{−2}(1 − \frac{ρ}{e^{it}} )^{−2} dt = \frac{1}{2πi} ∮_{T}(1 − ρz)^{−2}(1 − \frac{ρ}{z})^{−2} \frac{dz}{z} =

= ∑_{|z|<1} Res_{z} \frac{z}{(1 − ρz)^{2}(z − ρ)^{2}} = Res_{z=ρ} \frac{z}{(1 − ρz)^{2}(z − ρ)^{2}} =

= (\frac{z}{(1 − ρz)^{2}} )′|_{z=ρ} = \frac{1 + ρz}{(1 − ρz)^{3}} |_{z=ρ} = \frac{1 + Rr^{2}}{(1 − Rr^{2})^{3}} .

Hence

∫_{D} \frac{(1 − |x|^{2})^{a}}{|1 − \overline{y}x|^{4}} dν(x) = ∫_{0}^{1} \frac{1}{2π} ∫_{0}^{2π} \frac{(1 − r^{2})^{a}}{|1 − \overline{y}re^{it}|^{4}} dt 2r dr =
= ∫_{0}^{1} \frac{(1 − r^{2})^{a}(1 + Rr^{2})}{(1 − Rr^{2})^{3}} 2r dr =
= ∫_{0}^{1} \frac{(1 − t)^{a}(1 + Rt)}{(1 − Rt)^{3}} dt ≤
≤ 2 ∫_{0}^{1} \frac{(1 − t)^{a}}{(1 − Rt)^{3}} dt.

Find α ∈ (−1, 0) and β ∈ (0, 2) such that α + β = a ∈ (−1, 2). Then

\frac{(1 − t)^{a}}{(1 − Rt)^{3}} = (1 − R)^{β−2} (\frac{1 − R}{1 − Rt})^{2−β} (\frac{1 − t}{1 − Rt})^{β} \frac{(1 − t)^{α}}{1 − Rt} ≤ (1 − R)^{β−2} \frac{(1 − t)^{α}}{1 − Rt},

as \frac{1 − t}{1 − Rt}, \frac{1 − R}{1 − Rt} ∈ ⟨0, 1⟩.Owing to part (a),

∫_{0}^{1} \frac{(1 − t)^{α}}{1 − Rt} dt ≤ \frac{1}{(−α)(α + 1)} (1 − R)^{α}.

Hence

2 ∫_{0}^{1} \frac{(1 − t)^{a}}{(1 − Rt)^{3}} dt ≤ 2 (1 − R)^{β−2} . \frac{1}{(−α)(α + 1)} (1 − R)^{α} = C_{a} . (1 − R)^{a−2}

as asserted. □

Theorem 5.14. The Berezin transform B is a bounded operator on the spaces L^{p}(D, dν), 1 < p < ∞. Proof. Use the Schur test (Proposition 5.12) with P (x) = (1 − |x|^{2})^{−1/p}, Q(y) = (1 − |y|^{2})^{−1/q}:

∫_{D} \frac{(1 − |x|^{2})^{2}}{|1 − \overline{x}y|^{4}} (1 − |y|^{2})^{−1/p} dν(y) = (1 − |x|^{2})^{2} ∫_{D} \frac{(1 − |y|^{2})^{−1/p}}{|1 − \overline{x}y|^{4}} dν(y) ≤

≤ (1 − |x|^{2})^{2} . C_{−1/p} . (1 − |x|^{2})^{− \frac{1}{p} −2} = C_{−1/p} . P (x),

∫_{D} \frac{(1 − |x|^{2})^{2}}{|1 − \overline{x}y|^{4}} (1 − |x|^{2})^{−1/q} dν(x) ≤ C_{2− \frac{1}{q}} . (1 − |y|^{2})^{2− \frac{1}{q} −2} = C_{2− \frac{1}{q}} . Q(y),

by virtue of Lemma 5.13 □

Remark 5.15. The bound for the norm of B on L^{p}(D, dν), given by the Schur test (Proposition 5.12), is (C_{−1/p})^{1/q}.(C_{2− \frac{1}{q}})^{1/p}. Computing this explicitly leads to \frac{2p\sqrt{p}}{\sqrt{p^{2} − 1}}. □

Let us turn now to the second topic: determination of all functions which are invariant under the Berezin transform.

Proposition 5.16. If a function f ∈ L^{1}(D, dν) or L^{1}(C^{N}, dμ) is harmonic, then f̃ = f. Proof. If f ∈ L^{1}(D, dν) is harmonic, then so is f ∘ ω_{−w}; by the mean value property,

f̃(w) = ∫_{D} f(ω_{−w}(x)) dν(x) = f(ω_{−w}(0)) = f(w).

The case f ∈ L^{1}(C^{N}) is similar. □

A natural question to ask is if there are other functions such that f̃ = f. The following two propositions suggest that the answer is probably negative.

Proposition 5.17. (a) If f ∈ L^{2}(D, dη) and f̃ = f, then f is harmonic.

(b) If f ∈ L^{2}(C^{N}, dz) and f̃ = f, then f is harmonic. Proof. We shall employ the formulas (38), (39).

(b) Fix a selfadjoint Laplace operator Δ on L^{2}(C^{N}, dz) and let E(λ) be its resolution of the identity^{16}. Assume that f̃ = f; by (39), this is equivalent to e^{Δ/2} f = f. Consequently

0 = ∥(e^{Δ/2} − I)f∥^{2} = ∫_{R} |e^{λ/2} − 1|^{2} d⟨E(λ)f, f⟩.

16Continuity from the right is assumed It follows that e^{λ/2} = 1 d⟨E(λ)f, f⟩-almost everywhere on R — in other words, f belongs to the range of the projection E(0) − E(0−), which is exactly kerΔ. Thus Δf = 0 and we are done.

(a) is similar, only with the function F (λ) from (38) instead of e^{λ/2}. The formula

F (x)^{−1} = \frac{sin π (\sqrt{x + \frac{1}{4}} − \frac{1}{2})}{πx}

implies that, for x ∈ R, F (x) = 1 if and only if x = 0. Finally, Δ_{h}f = 0 is equivalent to Δf = 0. □

Remark 5.18. The last proposition does not say too much, since the only harmonic function in L^{2}(D, dη) or L^{2}(C^{N}, dz) is constant zero. To see this e.g. for L^{2}(D, dη), denote

M(r) = \frac{1}{2π} ∫_{0}^{2π} |f(re^{it})|^{2} dt.

This is a nonnegative and nondecreasing function of r. At the same time,

∥f∥_{L^{2} (D,dη)}^{2} = ∫_{0}^{1} M(r) . \frac{2r}{(1 − r^{2})^{2}} dr < +∞,

so M(r) must tend to zero as r → 1. Thus M(r) ≡ 0, whence f = 0. □

Therefore it would be desirable to extend the last result to some larger space — say, at least, to L^{∞}(D) or L^{∞}(C^{N}), respectively. In the Fock space setting, this can be done quite easily.

Proposition 5.19. Assume f ∈ L^{∞}(C^{N}), f̃ = f. Then f is harmonic (and, consequently, constant).

Proof. f = f̃ is locally integrable, and so determines a distribution on C^{N}; since f = f̃ is moreover bounded, this distribution is tempered, and we may apply the Fourier transform F. According to the definition of f̃, f̃ = e^{−|z|^{2} /2} ∗ f (convolution), so f̃ = f implies

e^{−|z|^{2} /2}.u(z) = u(z)

where u := Ff. In other words,

⟨u, (e^{−|z|^{2} /2} − 1)ϕ(z)⟩ = 0

for every ϕ ∈ S(C^{N}), the space of rapidly decreasing functions on C^{N} ≃ R^{2N}. Since e^{t} − 1 behaves like t when t → 0, the last condition is equivalent to

⟨u, −|z|^{2}ϕ(z)⟩ = 0 ∀ϕ ∈ S(C^{N}),

i.e. to −|z|^{2}u(z) = 0. Applying F^{−1} gives Δf = 0 as desired. □

This proof cannot be carried over verbatim to D, since there is no analogue of the Fourier transform which would behave reasonably with respect to Δ_{h}. Using the method from the proof of Theorem 5.22 below, it is possible to show that f ∈ L^{1}(D, dη) is harmonic if f̃ = f and Δf is Lebesgue integrable; the last condition, however, need not be a priori satisfied even for a bounded f. All that can be said is that Δ_{h}f must be bounded if f = f̃ ∈ L^{∞}(D):

Proposition 5.20. Assume that f ∈ L^{∞}(D). Then ∥Δ_{h}f̃∥_{∞} ≤ 5∥f∥_{∞}.

Proof. As usual, if w = x + yi ∈ D, denote \frac{∂}{∂w} = \frac{1}{2} (\frac{∂}{∂x} − i\frac{∂}{∂y} ) and \frac{∂}{∂\overline{w}} = \frac{1}{2} (\frac{∂}{∂x} + i\frac{∂}{∂y} ) . A short

computation yields

\frac{∂}{∂w} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} = \frac{2(\overline{z} − \overline{w})(1 − \overline{w}z)(1 − |w|^{2})}{|1 − \overline{w}z|^{6}}.

Because |z − w|.|1 − \overline{w}z| = |1 − \overline{w}z|^{2}. |\frac{z − w}{1 − \overline{w}z} | ≤ |1 − \overline{w}z|^{2}, it follows that

|f(z) . \frac{∂}{∂w} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} | ≤ ∥f∥_{∞} . \frac{2(1 − |w|^{2})}{|1 − \overline{w}z|^{4}} ≤ \frac{2∥f∥_{∞}}{(1 − |w|)^{4}} .

This is uniformly bounded when w runs through a small vicinity of an arbitrary point w_{0} ∈ D. Consequently, it is legal to differentiate under the integral sign in the formula defining f̃, which gives

\frac{∂f̃}{∂w} = ∫_{D} f(z) . \frac{2(\overline{z} − \overline{w})(1 − \overline{w}z)(1 − |w|^{2})}{|1 − \overline{w}z|^{6}} dν(z).

Going through the same procedure once again, we see that

\frac{∂^{2} f̃}{∂w∂\overline{w}} = ∫_{D} K(w, z)f(z) dν(z),

where the kernel is

K(z, w) = \frac{∂^{2}}{∂w∂\overline{w}} \frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} =
= (4|w|^{2} − 2)\frac{(1 − |w|^{2})^{2}}{|1 − \overline{w}z|^{4}} + 4\frac{1 − |w|^{2}}{|1 − \overline{w}z|^{4}} − 4\frac{(1 − |z|^{2})(1 − |w|^{2})}{|1 − \overline{w}z|^{6}}.

Recalling the formulas

(1 − \overline{w}z)^{−2} = ∑_{n=0}^{∞}(n + 1)\overline{w}^{n}z^{n} and (1 − \overline{w}z)^{−3} = ∑_{n=0}^{∞} \frac{(n + 1)(n + 2)}{2} \overline{w}^{n}z^{n}

and integrating term by term^{17}, we get

∫_{D} \frac{dν(z)}{|1 − \overline{w}z|^{4}} = (1 − |w|^{2})^{−2}

and

4 ∫_{D} \frac{1 − |z|^{2}}{|1 − \overline{w}z|^{6}} dν(z) = 2(1 − |w|^{2})^{−3},

respectively. Consequently,

|\frac{∂^{2} f̃}{∂w∂\overline{w}} | ≤ ∥f∥_{∞} . [(4|w|^{2} − 2) + \frac{4}{1 − |w|^{2}} + \frac{2}{(1 − |w|^{2})^{2}} ] ,

whence

|Δ_{h}f̃(w)| ≤ ∥f∥_{∞} . [(4|w|^{2} − 2)(1 − |w|^{2})^{2} + 4(1 − |w|^{2}) + 2].

For w ∈ D, the maximum of the bracketed term is approximately 4.439 … < 5; thus

∥Δ_{h}f̃∥_{∞} ≤ 5∥f∥_{∞}

as claimed. □

17This is legal (since the series converge uniformly on compact subsets of D) if we integrate over RD, 0 < R < 1; doing so and taking limits for R ↗ 1 leads to the desired formulas.

Using this proposition, a proof may be given that f is harmonic whenever f̃ = f ∈ L^{∞}(D). Since a lot technicialities seems to be necessary, only its sketch will be outlined here. Denote

M = {f ∈ L^{∞}(D) : f = f̃}.

This is a closed subspace of L^{∞}(D). The last proposition shows that A := Δ_{h}↾M is a bounded operator on M, ∥A∥ ≤ 5. Since

G(z) := ∏_{n=1}^{∞} (1 − \frac{z}{n(n + 1)} ) = \frac{sinπ (\sqrt{z + \frac{1}{4}} − \frac{1}{2})}{πz}

is an entire function, it is possible to define G(A) and, owing to the spectral mapping theorem,

σ(G(A)) = G(σ(A)).

Assuming^{18} that the formula (38) is valid for f and thinking for some time leads to the conclusion that G(Δ_{h}), whenever it is defined, must be the inverse of the Berezin transform. Consequently, G(A) = I and G(σ(A)) = {1}. Next, show that z = 0 is the only zero of G(z)−1 in the disc |z| ≤ 5; this implies σ(A) = {0}.

Let H(z) := \frac{G(z) − 1}{z} and C := H(A). C is clearly bounded; σ(C) = \{G′(0)\} = \{−1\}, so C is invertible; finally,

CA = G(A) − I = 0,

so A = 0, i.e. M ⊂ kerΔ_{h}. □

The last topic we wanted to discuss were ergodicity properties of the Berezin transform. Once again, situation is quite transparent when we restrict our attention to L^{2}(C^{N}, dz) and L^{2}(D, dη), but gets complicated if we want to deal with wider function classes.

Proposition 5.21. As an operator on L^{2}(C^{N}, dz) or L^{2}(D, dη), B^{n} → 0 in the strong operator topology.

Proof. We shall deal only with C^{N}, the proof for D being similar. Since B is a contraction of the form B = e^{Δ/2} for a selfadjoint operator Δ, its spectrum must be contained in ⟨0, 1⟩. Denote, once again, E(λ) the resolution of the identity for the (selfadjoint) operator B. Then

∥B^{n}f∥^{2} = ∫_{⟨0,1⟩} |λ^{n}|^{2} d⟨E(λ)f, f⟩.

According to the Lebesgue monotone convergence theorem, this tends to

∥I − E(1−))f∥^{2} = ∥P_{ker(B−I)}f∥^{2}.

But it follows from Proposition 5.17 and Remark 5.18 that ker(B −I) = {0}, whence ∥B^{n}f∥ → 0 as claimed. □

It is easy to see that this simple behaviour does not persist when we consider B on L^{∞}(C^{N}) or L^{∞}(D). For example, take f ∈ L^{∞}(C),

f(z) = \{ \begin{matrix}1 & if Rez > 0, \\ −1 & if Rez < 0. \end{matrix}

A short computation shows that

(B^{n}f)(z) = f̃(z/\sqrt{n}),

so (B^{n}f)(z) → f̃(0) = 0 as n → ∞ for all z ∈ C. However, the convergence cannot be uniform since, for every n ∈ N,

lim_{x→+∞} (B^{n}f)(x + yi) = lim_{x→+∞} f̃(x + yi) = +1.

18It seems necessary to resort to the theory of pseudodifferential operators in order to prove this.

All the same, the uniform convergence can be established in some particular cases. If f is harmonic, then Bf = f (Proposition 5.16), so trivially B^{n}f⇉f. What about subharmonic f?

Theorem 5.22. Assume that f ∈ L^{1}(D, dν) is a real-valued subharmonic function on D which admits an integrable harmonic majorant (i.e. there exists a function v ∈ L^{1}(D, dν) harmonic on D and such that v(x) ≥ f(x) ∀x ∈ D). Then B^{n}f ↗ u, the least harmonic majorant of f.

Proof. According to a theorem of Frostman ([19], Theorem 5.25), there exists a positive Borel measure κ on D such that

f(w) = u(w) + \frac{1}{4} ∫_{D} ln|ω_{w}(x)|^{2} dκ(x) ∀w ∈ D.

(When f is twice continuously differentiable, dκ(x) = Δf(x) dν(x); π dκ is called the Riesz measure of f.) Write, for brevity, g(x) = ln|x|^{2}. Since |ω_{w}(x)| = |ω_{x}(w)|, we have

f(w) = u(w) + \frac{1}{4} ∫_{D} g ∘ ω_{x}(w)dκ(x).

Hence

f̃(z) = ũ(z) + ∫_{D} \frac{(1 − |z|^{2})^{2}}{|1 − \overline{w}z|^{4}} . \frac{1}{4} ∫_{D} g ∘ ω_{x}(w)dκ(x) dν(w).

Since f ≤ u ≤ v and f, v ∈ L^{1}(D, dν), we have u ∈ L^{1}(D, dν), so ũ = u in view of Proposition 5.16. Because the integrand is nonpositive, we may interchange the order of integration, which gives

(42)  f̃(z) = u(z) + \frac{1}{4} ∫_{D} B(g ∘ ω_{x})(z) dκ(x).

Proceeding by induction, we obtain

(B^{n}f)(z) = u(z) + \frac{1}{4} ∫_{D} B^{n}(g ∘ ω_{x})(z) dκ(x).

Assume that

(43)  B^{n}(g ∘ ω_{x})(z) ↗ 0  as n → ∞, for all x, z ∈ D.

Because κ is a positive measure, we may apply the Lebesgue monotone convergence theorem to conclude that

B^{n}f ↗ u as n → ∞

and the proof of the theorem is complete.

It remains to prove (43). Since B^{n}(g ∘ ω_{x}) = (B^{n}g) ∘ ω_{x} (cf. Proposition 5.7), it suffices to show that B^{n}g ↗ 0. By definition, g(x) = ln|x|^{2}, while direct computation reveals that

g̃(x) = |x|^{2} − 1.

It follows that g ≤ g̃. By Proposition 5.4, this implies B^{k}g ≤ B^{k+1} g ∀k ∈ N, so

g ≤ Bg ≤ B^{2}g ≤ B^{3}g ≤ … ≤ 0.

Hence a limit ψ(x) := lim_{n→∞}(B^{n}g)(x) must exist, ψ ≤ 0, and, owing to the Lebesgue monotone convergence

theorem, ψ̃ = ψ. We claim that ψ ≡ 0. Assume the contrary. Because 0 ≥ ψ(x) ≥ g̃(x) = |x|^{2} − 1, we have lim_{|x|→1} ψ(x) = 0; consequently, ψ must attain its infimum at some point y ∈ D — suppose (replacing ψ by

ψ ∘ ω_{y} otherwise) that y = 0. Then

ψ(0) = ψ̃(0) = ∫_{D} ψ(x) dν(x) > inf_{D} ψ(x) . ∫_{D} dν(x) = ψ(0)

— a contradiction. □

Remark 5.23. If f is real-valued, subharmonic and f ∈ L^{2}(D, dη), we may proceed a little more quickly. The subharmonicity implies that

f̃(w) = ∫_{D} f(ω_{−w}(y)) dν(y) ≥ f(ω_{−w}(0)) = f(w),

i.e. f̃ ≥ f. Further, B commutes with Δ_{h}, whence Δ_{h}f̃ = B(Δ_{h}f) ≥ 0 since Δ_{h}f ≥ 0; in other words, f̃ is also subharmonic. Proceeding by induction, we obtain a nondecreasing sequence {B^{k}f}_{k∈N} of subharmonic functions. Their limit ψ is either identically +∞, or is a subharmonic function satisfying ψ̃ = ψ. Since ψ ∈ L^{2}(D, dη), the former case cannot occur; further,

ψ(0) = ψ̃(0) = ∫_{D} ψ(y) dν(y),

and so ψ is actually harmonic; hence, it is a harmonic majorant of f. If h is another harmonic majorant of f, then f ≤ h implies B^{n}f ≤ B^{n}h = h, whence also ψ ≤ h; consequently, ψ is the least harmonic majorant of f, and we are done.

Observe that, although there is no nonzero harmonic function in L^{2}(D, dη), there are plenty subharmonic ones. The functions B^{n}g, g(z) = ln|z|^{2}, n ∈ N, serve as an example:

∫_{D} |g(x)|^{2} dν(x) = ∫_{0}^{1} (\frac{lnt}{1 − t})^{2} dt = ∫_{0}^{1} ∑_{n=0}^{∞} ∑_{m=0}^{∞} t^{m+n} ln^{2} t dt =

= ∑_{n=0}^{∞} ∑_{m=0}^{∞} \frac{2}{(m + n + 1)^{3}} = ∑_{k=0}^{∞} \frac{2}{(k + 1)^{2}} = \frac{π^{2}}{3} < +∞.

□

In fact, another proof of Theorem 5.22 may be given on the lines of Remark 5.23 — the assumption f ∈ L^{2}(D, dν) was used only to infer that Bf is subharmonic when f is. Using the formula (42), this fact may be shown to hold in general.

Theorem 5.24. Assume f ∈ L^{1}(D, dν) is a real-valued subharmonic function on D which admits an integrable harmonic majorant v. Then the functions B^{n}f are subharmonic, ∀n ∈ N.

Proof. Let 0 < R < 1. Owing to the formula (42),

\frac{1}{2π} ∫_{0}^{2π} f̃(Re^{it}) dt = \frac{1}{2π} ∫_{0}^{2π} u(Re^{it}) dt + \frac{1}{2π} ∫_{0}^{2π} \frac{1}{4} ∫_{D} B(g ∘ ω_{x})(Re^{it}) dκ(x) dt.

Because the second integrand is nonpositive, we may interchange the order of integration; consequently,

\frac{1}{2π} ∫_{0}^{2π} f̃(Re^{it}) dt = u(0) + \frac{1}{4} ∫_{D} (\frac{1}{2π} ∫_{0}^{2π} B(g ∘ ω_{x})(Re^{it}) dt) dκ(x).

The function B(g ∘ ω_{x})(z) = |ω_{x}(z)|^{2} − 1 is a subharmonic function (of z), which implies

\frac{1}{2π} ∫_{0}^{2π} B(g ∘ ω_{x})(Re^{it}) dt ≥ B(g ∘ ω_{x})(0).

Hence

\frac{1}{2π} ∫_{0}^{2π} f̃(Re^{it}) dt ≥ u(0) + \frac{1}{4} ∫_{D} B(g ∘ ω_{x})(0) dκ(x) = f̃(0)

for every R ∈ (0, 1). The same procedure, of course, may be carried out for the functions f ∘ ω_{a}, a ∈ D. It follows that f̃ satisfies the sub-mean value property, and therefore is subharmonic on D.

Because f ≤ v, we have also f̃ ≤ ṽ = v (Propositions 5.4 and 5.16), so f̃ also has an integrable harmonic majorant. Consequently, we may proceed by induction, and the theorem follows. □

Given a bounded real-valued subharmonic function on D, the boundary values of its least harmonic majorant can be described explicitly. A proof of this fact is included below, since it seems to be missing in textbooks on potential theory.

Proposition 5.25. Suppose ϕ is a bounded real-valued subharmonic function on D. Define ϕ on T by

(44)  ϕ(ϵ) = lim sup_{r↗1}ϕ(rϵ),  ϵ ∈ T,

and let ψ be the Poisson extension of ϕ↾T into the interior of D. Then ψ is the least harmonic majorant of ϕ.

Proof. Let u be the least harmonic majorant of ϕ. Except for ϵ in a set of zero (arc-length) measure, we have

lim_{r↗1} u(rϵ) ≥ lim sup_{r↗1}ϕ(rϵ) = ϕ(ϵ) = lim_{r↗1} ψ(rϵ).

It follows that the bounded harmonic function u − ψ has nonnegative radial limits a.e. on T; hence, u ≥ ψ on D. Let us show that also u ≤ ψ. Because (sub)harmonicity is invariant under Möbius transformations, it suffices to show that ψ(0) ≥ u(0). Without loss of generality, we may assume ϕ ≤ 0; hence also u ≤ 0, ψ ≤ 0. Applying the Fatou lemma to the functions t ↦ ϕ(re^{it}), we see that

\frac{1}{2π} ∫_{0}^{2π} lim sup_{r↗1}ϕ(re^{it}) dt ≥ lim sup_{r↗1} \frac{1}{2π} ∫_{0}^{2π} ϕ(re^{it}) dt.

The left-hand side is, by definition, ψ(0), whereas the lim sup on the right-hand side may be replaced either by lim or by sup and equals u(0). □

Remark 5.26. The last proposition may be compared with Theorem 3.11 of [19], which asserts the same, but with (44) replaced by

ϕ(ϵ) = lim sup_{D∋z→ϵ}ϕ(z)

The proof of our version seems to be more elementary. □

The proofs of Theorem 5.22 as well as of Remark 5.23 may be carried over to the Fock space setting.

Theorem 5.27. If f is a bounded real-valued subharmonic function on C^{N}, then B^{n}f ↗ u, the least harmonic majorant of f.

Since the only bounded harmonic functions on C^{N} are the constant ones, necessarily

u(z) = sup_{x∈C^{N}} f(x) ∀z ∈ C^{N}.

As a matter of fact, an analogue of Proposition 5.25 remains in force: the above supremum coincides with the radial limit of f on almost all half-lines emanating from the origin^{19}.

Remark 5.28. For N = 1, the statement of the theorem is trivial, since there are no bounded subharmonic functions on C but constant ones. However, nontrivial bounded subharmonic functions exist when N ≥ 2; as an example, take max{−1, |z|^{2−2N} }. □

We conclude this chapter with an application of Theorems 5.22 and 5.27 Recall that

V (D) := {f ∈ L^{∞}(D) : ess lim_{|z|↗1}f(z) = 0}

19Cf. [19], Theorem 3.21. and similarly for C^{N}.

Proposition 5.29. If (a) f ∈ V (D) or (b) f ∈ V (C^{N}), then B^{n}f⇉0.

Proof. (a) First observe that it suffices to consider f ≤ 0 since B is linear. Next, it suffices to consider f ∈ D(D), because B is a contraction and D(D) is dense in V (D). So suppose f ≤ 0 and suppf ⊂ RD, R ∈ (0, 1). Define the function F on ⟨0, 1⟩ as follows:

F (t) = −∥f∥_{∞} if 0 ≤ t ≤ R,
F (1) = 0,
F (t) is linear on ⟨R, 1⟩,

and set ϕ(z) = F (|z|), z ∈ \overline{D}. The function ϕ is subharmonic, its least harmonic majorant being constant zero. By virtue of Theorem 5.22, B^{n}ϕ ↗ 0; since ϕ = 0 on T, Dini’s theorem forces even B^{n}ϕ⇉0. But ϕ ≤ f ≤ 0, hence B^{k}ϕ ≤ B^{k}f ≤ 0, and so B^{n}f⇉0 as well.

(b) The proof is easier this time, since an explicit formula for B^{n}f is well-known from the theory of the heat equation:

(B^{n}f)(z) = (2πn)^{−N/2} . ∫_{C^{N}} exp(−\frac{|z − x|^{2}}{2n} ) . f(x) dx.

Reasoning as above, it suffices to consider f ≥ 0 and suppf ⊂ RD, R ∈ (0, +∞). In that case,

|(B^{n}f)(z)| ≤ \frac{∥f∥_{∞}}{(2πn)^{N/2}} . ∫_{|x|≤R} dx = \frac{γ_{N}R^{2N}∥f∥_{∞}}{(2πn)^{N/2}} ∀z ∈ C^{N},

γ_{N} being the volume of the unit ball in C^{N}. Letting n → ∞ yields ∥B^{n}f∥_{∞} → 0 as claimed.

Corollary 5.30. Suppose f ∈ C(\overline{D}). Then B^{n}f⇉h, the harmonic function whose boundary values coincide with f↾T.

Proof. Because f ∈ C(\overline{D}), f↾T ∈ C(T), hence also h ∈ C(\overline{D}) and f − h ∈ V (D). It follows that B^{n}(f − h)⇉0. But Bh = h, so B^{n}f⇉h.

CONCLUSIONS FOR FURTHER DEVELOPMENT IN THE AREA

The results above lead to some challenging questions. It is still unclear what is the norm closure of the set of all Toeplitz operators on a Bergman-type space (Chapter 2); likewise, it is an open problem whether the C^{∗}-algebra generated by them is all of A or not, and whether A(U) and A(S) are isomorphic. One would also like to generalize Theorem 3.10 (or 3.14) to domains of higher dimensions; perhaps A(U) should play the ”universal” role which A(S) plays for Ω ⊂ C (cf. Remark 3.26). Many interesting problems are also encountered in Chapter 4 (the question of existence of some kind of Toeplitz calculus still remains open in general) and Chapter 5 (fixed points and ergodicity properties of the Berezin transform acting on L^{∞}, for example; Proposition 5.20 looks promising from this point of view).

LIST OF NOTATION

The numbers etc. refer to the place where the symbol is defined. Since there are paragraphs which are not numbered, references like ”3.30 & before” are used to indicate that the symbol first occurs between the items 3.29 and 3.30.

A^{p}(Ω) the Bergman (Ω ⊂ C^{N}) or the Fock (Ω = C^{N}) space A(M) the A-algebra corresponding to an operator M (3.30 & before) A^{♯}(M) the A^{♯}-algebra corresponding to M (3.30 & before)

A an abbreviation for A(S) (3.4) A_{μ} the localization of A at μ ∈ T (4.16) A_{ab} = A/ComA

B the Berezin transform, Bf ≡ f̃ B_{ϕ} = W_{B}^{∗}T_{ϕ}W_{B} (the very end of Chapter 3)

BCESV a function space, see 3.25 & before B(H) the space of all bounded linear operators on a Hilbert space H BMO the space of all functions of bounded mean oscillation (sometimes also called the

John-Nirenberg space, and denoted JN) C the complex plane

c_{n}(f) see 1.3, 1.7

Comp , Comp (H) the compact operators (on a Hilbert space H) closX closure of a set X ⊂ B(H) in the operator norm topology C(\overline{Ω} ) the space of functions continuous on the closure \overline{Ω} of a domain Ω ⊂ C^{N}

Calk, Calk(H) the Calkin algebra (on a Hilbert space H) ComA the commutator ideal of an algebra A

C^{∗}-algX the smallest C^{∗}-algebra containing a set X ⊂ B(H) C the adjoint of B (5.8) C_{a} a constant (5.13) d X^{d}, the dual of a Banach space X; T^{d}, the dual of an operator T D the unit disc dz the Lebesgue measure on C^{N}, N ≥ 1

dν(z) = \frac{1}{π} dz, a measure on D

dμ(z) the Gaussian measure on C^{N} dκ(z) the Riesz measure of a subharmonic function (5.21)

diag(c_{n}) the diagonal operator with weight sequence {c_{n}}

dist the (euclidean) distance d(x, y) the pseudohyperbolic distance of x, y ∈ D

D_{h}(λ, R) the pseudohyperbolic disc with center λ and radius R

D(Ω) the space of all infinitely differentiable functions the supports of which are compact

subsets of Ω ⊆ C^{N} ess lim the essential limit; similarly ess sup ESV a function space, see 3.25 & before {E_{N}} an orthonormal basis in H and K (4.8)

E, Ẽ spaces of operators, see 4.12 {e_{j}} orthonormal bases in various Hilbert spaces E(λ) resolution of the identity of a selfadjoint operator (continuity from the right is

assumed) F_{ϕ} = W_{F}^{∗}T_{ϕ}W_{F} (the very end of Chapter 3) F the Fourier transform g_{λ} the reproducing kernel (or evaluation functional) at λ, for various spaces (A^{2}(Ω) ,

A^{2}(C^{N}), H^{2}(ρ))

g_{(m,a)} the functional evaluating the m-th derivative at λ, on A^{2}(D) (before Lemma 2.6)

G the Gauss sphere, C ∪ {∞} g(x) = ln|x|^{2} at the end of Chapter 5

H^{∞}(Ω) the space of bounded analytic functions on Ω H_{ϕ} the Hankel operator with symbol ϕ H^{2} the Hardy space (on the unit circle) H_{−}^{2} = L^{2}(T) ⊖ H^{2}

H^{2}(ρ) Hilbert spaces, see before 3.18 H a subspace of L, see 4.8 iff if and only if ind the Fredholm index I_{μ} the local ideal, see 4.16 J 1) the diagonal operator J = diag(−1)^{n}

2) an isometry of H^{2} onto H, see 4.8 k_{λ} normalized evaluation functionals, k_{λ} = \frac{g_{λ}}{∥g_{λ}∥}

K a subspace of L, see 4.8 ff.

L^{p}, L^{p}(Ω), L^{p}(Ω, dρ) Lebesgue spaces (1 ≤ p ≤ ∞); if dρ is omitted, the Lebesgue measure is understood

L^{2}(ρ), L^{∞}(ρ) function spaces, see before 3.20

l^{2} = l^{2}(N) the space of all square-summable sequences of complex numbers, endowed with the

usual Hilbert space structure

l^{∞} = l^{∞}(N) the space of all bounded sequences of complex numbers, with supremum norm

L_{∞} a subset of Calk(L^{2}(T)), see 3.32 L′, L′′, L spaces used to construct the Calkin representation (see before 4.3) (exceptionally,

the primes do not denote commutants)

Lim a Banach limit, see Chapter 4 M_{ϕ} the operator of multiplication by a ϕ ∈ L^{∞} N = {0, 1, 2, …} n̂ the functional on l^{∞}: {x_{k}}_{k∈N} ↦ x_{n} P_{+} the orthogonal projection of L^{2} onto H^{2} or A^{2}, in various settings QC the quasicontinuous functions on T

RanT the range of a mapping T R the set of all real numbers R_{ϵ} a composition operator, see 4.17

supp the support (of a function or a measure) S the unilateral (forward) shift on H^{2} S the unit sphere in C^{N}

S(C^{N}) the Schwarz space of rapidly decreasing functions

SOT the strong operator topology 𝔖_{0}, 𝔖̃_{0} spaces of operators, see page 51 and before 4.11, respectively

T the unit circle in C T_{ϕ} the Toeplitz operator with symbol ϕ T = {T_{ϕ} : ϕ ∈ L^{∞}(D)}, a subset of B(A^{2}(D)) T_{1} = {T_{ϕ} : ϕ ∈ D(D)}, a subset of B(A^{2}(D)) T_{2} = \{T_{ϕ} : ϕ ∈ C(\overline{D})\}, a subset of B(A^{2}(D))

Trace the ideal of trace-class operators Tr(T) the trace of an operator T

T_{(m,n,a)} operators on A^{2}(D), see before 2.6 T_{B}, T_{F} , T_{H} the C^{∗}-algebras generated by all Toeplitz operators with bounded symbols on

A^{2}(D), A^{2}(C^{N}), H^{2} t_{a} the translation operator, t_{a}(z) = z − a, a, z ∈ C^{N} U the bilateral (forward) shift on L^{2}(T)

V (D) = {ϕ ∈ L^{∞}(D) : ess lim_{|z|↗1}ϕ(z) = 0}

V (C^{N}) = {ϕ ∈ L^{∞}(C^{N}) : ess lim_{|z|↗+∞} ϕ(z) = 0}

W OT the weak operator topology W_{B}, W_{F} the unitary operators of H^{2} onto A^{2}(D), A^{2}(C^{N}) (see the very end of Chapter 3)

X_{1}(D), X_{1}(0, 1) Banach function spaces, see 5.11 Z the set of all integers Z the (forward) shift operator on H^{2}(ρ), in 3.20 – 3.22; in 3.24, a (forward) shift

operator on A^{2}(C)

βN the Stone-Čech compactification of N γ_{N} the (Lebesgue) volume of the unit ball in C^{N} Γ(x) Euler’s gamma-function, n! = Γ(n + 1)

Δ the Laplace operator Δ_{h} the Laplace-Beltrami operator on D, see 5.7 dη an invariant measure on D, dη(z) = (1 − |z|^{2})^{2} dν(z) ϕ_{k} the k-th Fourier coefficient of ϕ ∈ L^{2}(T) Ω a domain in C^{N}, or C^{N} (see the Convention at the beginning of Chapter 1) |Ω| the Lebesgue measure of Ω

\overline{Ω} , (∂Ω) the closure (boundary) of Ω ⊂ C^{N} in G^{N}

π the canonical projection of B(H) onto Calk(H)

π_{n}(T), π(S, T) homotopy groups σ the spectrum σ_{e} the essential spectrum

τ_{B}, τ_{H}, τ_{F} the set of all Toeplitz operators with bounded symbol, on A^{2}(D), H^{2}, A^{2}(C^{N}),

respectively

ξ, ξ_{B}, ξ_{F} symbol maps on H^{2}, A^{2}(D), A^{2}(C^{N}), see page 35 & ff.

If n = (n_{1}, n_{2}, …, n_{N}) is a multiindex, then

a_{n} = a_{n_{1} ,n_{2} ,...,n_{N}} , n! = n_{1}!n_{2}! … n_{N}!,
2^{n} = 2^{n_{1} +n_{2} +...+n_{N}} , z^{n} = z_{1}^{n_{1}}z_{2}^{n_{2}} … z_{N}^{n_{N}} .

If x, y ∈ C^{N}, then

\overline{x}y = y\overline{x} = ∑_{j=1}^{N} \overline{x}_{j}y_{j}, |x| = (\overline{x}x)^{1/2}.

If ω = x + yi ∈ C, then

\frac{∂}{∂ω} = \frac{1}{2} (\frac{∂}{∂x} − i\frac{∂}{∂y} ) , \frac{∂}{∂\overline{ω}} = \frac{1}{2} (\frac{∂}{∂x} + i\frac{∂}{∂y} ) .

:= equals by definition [A, B] = AB − BA, the commutator of A and B [T_{f}, T_{g}) = T_{fg} − T_{f}T_{g}, the semicommutator

T^{♯} an operator on L, see before 4.4 ′ the first derivative, or the commutant (exception: L′)

′′ the double commutant (exception: L′′) T̂ see before 4.19

the Berezin transform, on L^{∞}(D), L^{∞}(C^{N}), B(A^{2}(D)),

B(A^{2}(C^{N}))

(k) the k-tuple iterate of ♭ a smoothing transformation, see before 1.11 1 the constant function equal to one

∥ . ∥_{B} norm on a Banach space B; B is frequently omitted ⟨ . , . ⟩_{H} scalar product on a Hilbert space H; H is sometimes omitted

∥ . ∥_{p} norm on L^{p} ∥ . ∥_{ρ}, ⟨ . , . ⟩_{ρ} norm & scalar product on H^{2}(ρ) ∥ . ∥_{L}, ⟨ . , . ⟩_{L} norm & scalar product on L

∥ . ∥_{X} norm in X_{1} ∥ . ∥_{e} the essential norm (of an operator)

∥ . ∥_{p→p}, ∥ . ∥_{L→L} norm of an operator from L^{p} into L^{p}, from L into L → converges ⇉ converges uniformly ↗ converges increasingly →^{w} converges weakly (on a Banach space)

∗ convolution ↾ restriction

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