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

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 262)

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.

Keywords

Noncommutative function completely positive noncommutative kernel noncommutative Schur–Agler class noncommutative contractive-multiplier class noncommutative Pick interpolation 

Mathematics Subject Classification (2010)

47B32 47A60 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • Gregory Marx
    • 1
  • Victor Vinnikov
    • 2
  1. 1.Department of MathematicsVirginia Tech BlacksburgUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael

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