Noncommutative reproducing kernel Hilbert spaces

Joseph A. Ball, Gregory Marx, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, free probability, and optimization theory in linear systems engineering). The purpose of this article is to develop a theory of positive kernels and associated reproducing kernel Hilbert spaces for the setting of free noncommutative function theory.

Original languageEnglish
Pages (from-to)1844-1920
Number of pages77
JournalJournal of Functional Analysis
Volume271
Issue number7
DOIs
StatePublished - 1 Oct 2016

Keywords

  • Completely positive and completely bounded maps
  • Contractive multiplier
  • Free noncommutative function
  • Reproducing kernel Hilbert space

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