Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Ball, Joseph A.; Marx, Gregory; Vinnikov, Victor

doi:10.1007/978-3-319-62527-0_3

Cited by 47 publications

(87 citation statements)

References 90 publications

(225 reference statements)

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“…They have obtained interpolation and realization results with applications to H ∞ functional calculus on free analogs of polynomial polyhedra. Similar interpolation and realization results were obtained by Ball, Marx and Vinnikov in [14] and [13], where they have also developed the theory of noncommutative reproducing kernels and defined the complete Pick property for such kernels. Muhly and Solel formulated a much more general theory using W * -correspondences in [47] (see also [49]).…”

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confidence: 72%

“…OF NONCOMMUTATIVE COMPLETE PICK SPACES Let d < ∞ and F d be the full Fock space on d generators. As shown in[13] F d is a noncommutative reproducing kernel Hilbert space (nc-RKHS for short) with the noncom-Here W d is the monoid of words on d letter and for Z ∈ B d , Z α is the evaluation of the monomial defined by the word α on Z. In[13] and[65] using different techniques, it was shown that K is a complete Pick kernel. Let us write H ∞ (B d ) for the algebra of bounded nc functions on B d .…”

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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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confidence: 72%

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confidence: 99%

On fixed points of self maps of the free ball

Shamovich

2018

Journal of Functional Analysis

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“…The case where τ is the free topology will be of special interest here, and in this case we refer to τ -holomorphic functions as free (or freely) holomorphic functions. Such functions are particularly well behaved on account of the following theorem [1] (it is also proved in [3,6]…”

Section: Holomorphy With Respect To Admissible Topologiesmentioning

confidence: 91%

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Agler

McCarthy

Young

2018

Trans. London Math. Soc.

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The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d‐dimensional nc‐universe Md. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.

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“…All three known proofs of Theorem 1.5 start by proving a realization on finite sets, and then somehow taking a limit. In [2], this is done by considering partial nc-functions; in [6], it is done by using noncommutative kernels to get a compact set in which limit points must exist; in the current paper, we use the wandering Montel theorem.…”

Section: To Provementioning

confidence: 99%