Test Functions, Schur-Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

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Abstract

Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with ℂN of the scalar-valued version.

Original languageEnglish
Pages (from-to)529-575
Number of pages47
JournalComplex Analysis and Operator Theory
Volume7
Issue number3
DOIs
StatePublished - 1 Jun 2013
Externally publishedYes

Keywords

  • Completely positive kernels
  • Constrained H-algebra
  • Finitely connected planar domain
  • Interior point of convex hull
  • Internal tensor product of correspondences
  • Positive kernels
  • Reproducing kernel Hilbert spaces
  • Schur-Agler class
  • Separation of convex sets
  • Test functions
  • Transfer-function realization
  • Unitary colligation matrix

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