Noncommutative reproducing kernel Hilbert spaces

Joseph A. Ball, Gregory Marx, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review


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 languageAmerican English
Pages (from-to)1844-1920
Number of pages77
JournalJournal of Functional Analysis
Issue number7
StatePublished - 1 Oct 2016


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

All Science Journal Classification (ASJC) codes

  • Analysis


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