AN ANALYTIC MODEL FOR COMMUTING

OPERATOR TUPLES

Miroslav Engliš

(a joint work with C. Ambrozie and V. Müller)

Let T be a bounded linear operator on a separable Hilbert space H. A well-known result of Sz.-Nagy and Foias says that the following two assertions are equivalent:

(a) I − T T^{∗} ≥ 0 (i.e. T is a contraction) and T^{∗n} → 0 in the strong operator topology;

(b) T^{∗} is unitarily equivalent to the restriction of a backward shift of infinite

multiplicity to an invariant subspace. In this talk we describe a generalization of this result for commuting n-tuples T = (T_{1}, …, T_{n}) of operators on H.

Let Ω be a domain in ℂ^{n} and H a Hilbert space of analytic functions on Ω which satisfies the following properties:

(1) H is invariant under the operators Z_{j} of multiplications by the coordinate functions (j = 1, …, n).

(2) The evaluation functionals are continuous on H. Consequently, there exists a reproducing kernel K(z, w) for H.

(3) H contains all polynomials and they are dense in it.

(4) The function 1/K(z, w) is a polynomial (in z and \overline{w}).

It is well known that for any orthonormal basis {ψ_{k}} of H, the reproducing kernel is given by

K(z, w) = ∑_{k} ψ_{k}(z)\overline{ψ_{k}(w)}.

In view of (3), by applying the Gramm-Schmidt orthogonalization process, we may construct a basis such that all ψ_{k} are polynomials (and, conversely, any polynomial is a linear combination of a finite number of the ψ_{k}). We fix such a basis from now on. For each m, set

f_{m}(z, \overline{w}) = ∑_{k≥m} \frac{ψ_{k}(z)\overline{ψ_{k}(w)}}{K(z, w)}.

Then f_{0}(z, \overline{w}) ≡ 1 on Ω × Ω. By virtue of (4) and our choice of the basis, the difference f_{0} − f_{m} is a polynomial in z, \overline{w}, for each m; thus f_{m} themselves are, in fact, polynomials.

Typeset by AMS-TEX

For any polynomial p(z, \overline{w}) of z, \overline{w} ∈ ℂ^{n}, define

p(T, T^{∗}) = ∑_{α,β} p_{αβ}T^{α}T^{∗β}  if  p(z, \overline{w}) = ∑_{α,β} z^{α}\overline{w}^{β}.

(Up to the order of T and T^{∗}, this coincides with the “hereditary calculus” of Agler [A1].) Hence, in particular, f_{m}(T, T^{∗}) are defined for any commuting operator tuple T, and f_{0}(T, T^{∗}) = I. Our main result is the following.

Theorem. Let H, ψ_{k}, f_{m} be as above and let T be a commuting n-tuple of operators on a separable Hilbert space H. Denote by M_{z} the operator n-tuple Z ⊗ I on the Hilbert space tensor product H ⊗ H. Then the following are equivalent:

(a) \frac{1}{K} (T, T^{∗}) ≥ 0 and ⟨f_{m}(T, T^{∗})h, h⟩ → 0 ∀h ∈ H.

(b) T^{∗} is unitarily equivalent to a restriction of M_{z}^{∗} to an invariant subspace.

Example. Let n = 1, Ω = 𝔻, the unit disc, and H = H^{2}, the Hardy space. Then K(z, w) = 1/(1−z\overline{w}), so \frac{1}{K} (T, T^{∗}) = I −T T^{∗}, and using the standard orthonormal basis ψ_{k}(z) = z^{k} we get f_{m}(T, T^{∗}) = T^{m}T^{∗m}. Further, the operator M_{z} on the tensor product H^{2}⊗H is just the forward shift of infinite multiplicity (its wandering subspace is 1 ⊗ H). Thus we recover the result of Nagy-Foias mentioned in the beginning. □

Other results covered by the last Theorem include the regular dilations of commuting n-tuples of operators (for H = the Hardy space on the polydisc 𝔻^{n}), the k-hypercontractions of Agler [A2] (for H the Bergman space on 𝔻 with respect to the weight (1 − |z|^{2})^{k−2}), and the “spherical hypercontractions” of Müller and Vasilescu [MV] (for H a certain weighted Bergman space on the unit ball of ℂ^{n}).

Sketch of the proof. (a) ⟹ (b). Denote D_{T} = \frac{1}{K} (T, T^{∗})^{1/2}. Define an operator V : H → H ⊗ H by

(*)  V h = ∑_{k} ψ_{k}(z) ⊗ D_{T} ψ_{k}(T)^{∗}h.

We claim that V is well-defined (i.e. the sum converges) and is, in fact, an isometry satisfying V T^{∗} = M_{z}^{∗}V. This clearly establishes (b).

To see that V is well-defined and an isometry, observe that for any j < m and h ∈ H

∥ ∑_{j≤k<m} ψ_{k}(z) ⊗ D_{T} ψ_{k}(T)^{∗}h∥^{2} = ∑_{j≤k<m} ∥D_{T} ψ_{k}(T)^{∗}h∥^{2}

= ∑_{j≤k<m}⟨ψ_{k}(T)D_{T}^{2} ψ_{k}(T)^{∗}h, h⟩

= ⟨(f_{j} − f_{m})(T, T^{∗})h, h⟩.

As ⟨f_{m}(T, T^{∗})h, h⟩ → 0 by hypothesis, it follows that the partial sums of the righthand side of (*) form a Cauchy sequence, and letting j = 0 and m → ∞ shows that ∥V h∥^{2} = ⟨f_{0}(T, T^{∗})h, h⟩ = ∥h∥^{2}, i.e. V is an isometry.

To prove that V T^{∗} = M_{z}^{∗}V, observe that ∀h, h′ ∈ H and any k

⟨V h, ψ_{k} ⊗ h′⟩ = ⟨D_{T} ψ_{k}(T)^{∗}h, h′⟩ = ⟨h, ψ_{k}(T)D_{T} h′⟩,

so, by virtue of our choice of the basis ψ_{k},

(†)  ⟨V h, f ⊗ h′⟩ = ⟨h, f(T)D_{T} h′⟩

for any polynomial f. Thus

⟨V T_{j}^{∗}h, ψ_{k} ⊗ h′⟩ = ⟨T_{j}^{∗}h, ψ_{k}(T)D_{T} h′⟩ = ⟨h, T_{j}ψ_{k}(T)D_{T} h′⟩ =

= ⟨h, (z_{j}ψ_{k})(T)D_{T} h′⟩ = ⟨V h, z_{j}ψ_{k} ⊗ h′⟩ = ⟨M_{z_{j}}^{∗}V h, ψ_{k} ⊗ h′⟩,

and the assertion follows.

(b) ⟹ (a). Let U : H → H ⊗ H be an isometry such that UT^{∗} = M_{z}^{∗}U. Then a simple calculation shows that for any polynomial p(z, \overline{w}) in z and \overline{w},

p(T, T^{∗}) = U^{∗}p(M_{z}, M_{z}^{∗})U = U^{∗}(p(Z, Z^{∗}) ⊗ I)U

(here, as before, Z stands for the operator tuple of multiplication by the coordinate functions on H). Thus it suffices to show that \frac{1}{K} (Z, Z^{∗}) ≥ 0 and ⟨f_{m}(Z, Z^{∗})h, h⟩ → 0 ∀h ∈ H.

To see the former, recall that for any w ∈ Ω

Z_{j}^{∗}K_{w} = \overline{w}_{j}K_{w}

where K_{w}(z) := K(z, w). It follows that for any polynomial p(z, \overline{w}) and x, y ∈ Ω,

⟨p(Z, Z^{∗})K_{y}, K_{x}⟩ = p(x, \overline{y})⟨K_{y}, K_{x}⟩ = p(x, \overline{y})K(x, y),

so, in particular, ⟨\frac{1}{K} (Z, Z^{∗})K_{y}, K_{x}⟩ = 1 ∀x, y ∈ Ω. As also ⟨K_{y}, 1⟩⟨1, K_{x}⟩ = \overline{1(y)}1(x) = 1 ∀x, y ∈ Ω, it follows that

(**)  \frac{1}{K} (Z, Z^{∗}) = ⟨·, 1⟩1

which is a positive operator.

For the latter assertion, observe that

(f_{0} − f_{m})(Z, Z^{∗})h = ∑_{0≤k<m} ψ_{k}(Z)\frac{1}{K} (Z, Z^{∗})ψ_{k}(Z)^{∗}h

= ∑_{0≤k<m} ψ_{k}(Z)(⟨·, 1⟩1)ψ_{k}(Z)^{∗}h by (**)

= ∑_{0≤k<m}⟨ψ_{k}(Z)^{∗}h, 1⟩ψ_{k}

= ∑_{0≤k<m}⟨h, ψ_{k}⟩ψ_{k},

and as ψ_{k} is an orthonormal basis and f_{0} ≡ 1, it follows that

(f_{0} − f_{m})(Z, Z^{∗})h = h − f_{m}(Z, Z^{∗})h → h as m → ∞,

i.e. even f_{m}(Z, Z^{∗})h → 0 ∀h ∈ H. This completes the proof. □

We remark that in view of the boundedness of V, the formula (†) defines a (nonmultiplicative) “functional calculus” g ↦ g(T) for functions g on Ω of the form g(z) = K(z, z)^{−1/2}f(z), f ∈ H (defining g(T)h ( =“f(T)D_{T} h”) := V^{∗}(f ⊗ h) one has ∥g(T)∥ ≤ ∥V ∥∥f∥).

An example of function spaces H satisfying the axioms (1) – (4) are, for instance, various weighted Bergman and Hardy spaces on bounded symmetric domains in ℂ^{n} (matrix balls etc.).

Under the additional hypothesis that the Taylor spectrum σ(T) ⊂ Ω, it turns out that the condition ⟨f_{m}(T, T^{∗})h, h⟩ → 0 can be omitted, and the axioms (3) and (4) for H replaced by the weaker requirement that K(z, w) ≠ 0 on Ω × Ω. For this and further details we refer to the joint work [AEM].

References

[A1] J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations and Operator Theory 5 (1982), 608–631.

[A2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 201–217.

[MV] V. Müller, F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), 979–989.

[AEM] C. Ambrozie, M. Engliš, V. Müller, Operator tuples and analytic models over general domains in ℂ^{n}, preprint (1999).