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In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...
We give a new constructive proof of the composition rule for Taylor's functional calculus for commuting operators on a Banach space.
Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in . The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.
Let T be a spherical 2-expansive m-tuple and let denote its spherical Cauchy dual. If is commuting then the inequality
holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.
We introduce the concept of analytic spectral radius for a family of operators indexed by some finite measure space. This spectral radius is compared with the algebraic and geometric spectral radii when the operators belong to some finite-dimensional solvable Lie algebra. We describe several situations when the three spectral radii coincide. These results extend well known facts concerning commuting n-tuples of operators.
We extend the notion of ascent and descent for an operator acting on a vector space to sets of operators. If the ascent and descent of a set are both finite then they must be equal and give rise to a canonical decomposition of the space. Algebras of operators, unions of sets and closures of sets are treated. As an application we construct a Browder joint spectrum for commuting tuples of bounded operators which is compact-valued and has the projection property.
Using axiomatic joint spectra we obtain a functional calculus which extends our previous Gelfand-Waelbroeck type results to include a Banach-valued Taylor-Waelbroeck spectrum.
The relationship between the joint spectrum γ(A) of an n-tuple of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators,...
* Partially supported by Grant MM-428/94 of MESC.In this paper we present some generalizations of results of M.
S. Livšic [4,6], concerning regular colligations (A1, A2, H, Φ, E, σ1, σ2, γ, ˜γ)
(σ1 > 0) of a pair of commuting nonselfadjoint operators A1, A2 with finite
dimensional imaginary parts, their complete characteristic functions and a
class Ω(σ1, σ2) of operator-functions W(x1, x2, z): E → E in the case of
an inner function W(1, 0, z) of the class Ω(σ1). ...
The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel for |z|,|w| < 1, by means of , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful...
Motivated by some structural properties of Drury-Arveson d-shift, we investigate a class of functions consisting of polynomials and completely monotone functions defined on the semi-group ℕ of non-negative integers, and its operator-theoretic counterpart which we refer to as the class of completely hypercontractive tuples of finite order. We obtain a Lévy-Khinchin type integral representation for the spherical generating tuples associated with such operator tuples and discuss its applications.
Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of ,
,
where Sp denotes the joint Taylor spectrum.
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