2018
DOI: 10.1007/978-3-319-62527-0_3
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Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

Abstract: The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variable… Show more

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Cited by 47 publications
(87 citation 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]).…”
supporting
confidence: 72%
See 1 more Smart Citation
“…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]).…”
supporting
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 .…”
mentioning
confidence: 99%
“…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%
“…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%