Interpolation and transfer-function realization for the noncommutative schur–agler class

Joseph A. Ball, Gregory Marx, Victor Vinnikov

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

12 Scopus citations

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 variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur–Agler-class theory to the free noncommutative function setting. This includes the positive-kerneldecomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur–Agler class coincides with the contractivemultiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.

Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Pages23-116
Number of pages94
DOIs
StatePublished - 1 Jan 2018

Publication series

NameOperator Theory: Advances and Applications
Volume262
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Completely positive noncommutative kernel
  • Noncommutative contractive-multiplier class
  • Noncommutative function
  • Noncommutative pick interpolation
  • Noncommutative schur–agler class

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