Non-commutative holomorphic functions on operator domains

Agler, Jim; McCarthy, John E.

doi:10.1007/s40879-015-0064-2

Cited by 30 publications

(27 citation statements)

References 21 publications

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“…An aim of operatorial free function theory is to understand functional calculus for several noncommuting operators and its applications. (See e. g. [4,5,32,15,29,26].) We will see that invariant structure preserving functions are in the pointwise weak closure of the polynomials and therefore free.…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 89%

“…A result of Agler-McCarthy and Ball-Marx-Vinnikov [4,9] states that whenever there is a matrix input that makes δ contractive, then B δ satisfies the above theorem.…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 99%

“…We leave the details to the reader. For free functions on large enough domains, including polynomially convex domains, or more generally open "free sets," locally bounded (in the norm topology) functions must be analytic [4].…”

Section: 3mentioning

confidence: 99%

“…Note that we require no hypotheses on the domain containing 0 or being connected as was the case in [4].…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 99%

“…(4) A contractive free noncommutative function on a polynomially convex domain can be approximated by contractive polynomials. Importantly, we eliminate the hypothesis that the domain contain a scalar or matricial point as was assumed in [4,9]. See Theorem 5.6.…”

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Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Pascoe

2021

Preprint

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We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of polynomials in the strong topology and therefore a free function. Moreover, if a domain of operators on a Hilbert space is polynomially convex, the set of free functions satisfies a Oka-Weil Kaplansky density type theorem-contractive functions can be approximated by contractive polynomials.

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Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 89%

“…A result of Agler-McCarthy and Ball-Marx-Vinnikov [4,9] states that whenever there is a matrix input that makes δ contractive, then B δ satisfies the above theorem.…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…Note that we require no hypotheses on the domain containing 0 or being connected as was the case in [4].…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 99%

mentioning

confidence: 99%

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Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Pascoe

2021

Preprint

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Function Theory from Tensor Algebras

Muhly

Solel²

2022

Complex Anal. Oper. Theory

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On fixed points of self maps of the free ball

Shamovich

2018

Journal of Functional Analysis

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In this paper, we study the structure of the fixed point sets of noncommutative self maps of the free ball. We show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace. We provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete Pick spaces. 1 1− z,w (see [11] and [24]). This space is a complete Pick space, i.e., the multipliers of the Drury-Arveson space admit an interpolation theorem for matrix valued functions generalizing the classical Nevanlina-Pick interpolation theorem in the unit disc. Let us write M d for the algebra of multipliers on H 2 d , it is a maximal abelian WOT-closed operator subalgebra of B(H 2 d ) generated by the operators M z j of multiplication by coordinate functions. In [3, Theorem 8.2] it was shown that the Drury-Arveson space for d = ∞ is the universal complete Pick space, namely, if H is a separable complete Pick reproducing kernel Hilbert space on a set X with kernel k, then there exists an embedding b : X → B ∞ and a nowhere vanishing function δ on X, such that k(x, y) = δ(x)δ(y)k ∞ (b(x), b(y)) and H is isometrically embedded in δH 2 d . Let V ⊂ B d be an analytic subvariety of B d cut out by functions in M d . We can associate to it a reproducing kernel Hilbert space H V spanned by kernel functions k d (·, w) for w ∈ V . This space turns out to be a complete Pick space and the multiplier algebraof functions vanishing on V . It is thus natural to ask to what extent does the algebra M V determine the variety V and vice versa. The isomorphism problem for subvarieties of B d cut out by multipliers of the Drury-Arveson space was studied by Davidson, Ramsey and Shalit. In [22] and [23] they studied the algebra M V and its norm closed analog and proved that if V, W ⊂ B d are subvarieties of B d , such that their affine span is all of C d , then M V is completely isometrically isomorphic to M W if and only if there is an automorphism of B d mapping V onto W (see also [64] for a survey and more results on the commutative isomorphism problem). One of the main tools in the proof of the theorem is a theorem that appears both in [63] and [39] and states that the fixed point set of a self map of B d is the arXiv:1709.00446v2 [math.OA] 24 Dec 2018 intersection of B d with an affine subspace (see also [32, Theorem 23.2] and [42, Theorem 6.3]). The noncommutative (nc for short), or free functions were introduced by Taylor in [66] and [67]. Taylor's goal was to facilitate noncommutative functional calculus and thus he endeavored to give topological algebras analogous to the classical Frechet algebras of analytic functions on open domains in C d . Voiculescu in [72], [73], [74] and [75] developed the ideas of Taylor in the context of free probability. Helton, Klep , McCullough and Schweighofer applied noncommutative analysis in order to obtain dimension free relaxation of the LMI containment problem (see [34] and [35]). Their results were extended and impr...

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Non-commutative holomorphic functions on operator domains

Abstract: We characterize functions of d-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.

Cited by 30 publications

References 21 publications

Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Function Theory from Tensor Algebras

On fixed points of self maps of the free ball

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