Noncommutative reproducing kernel Hilbert spaces

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 func...

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Bibliographic Details
Published inJournal of functional analysis Vol. 271; no. 7; pp. 1844 - 1920
Main Authors Ball, Joseph A., Marx, Gregory, Vinnikov, Victor
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2016
Subjects
Completely positive and completely bounded maps
Contractive multiplier
Free noncommutative function
Reproducing kernel Hilbert space
47B32
Contractive multiplier
Reproducing kernel Hilbert space
Completely positive and completely bounded maps
47A60
Free noncommutative function
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Summary: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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2016.06.010