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arXiv · 1602.00760

Noncommutative reproducing kernel Hilbert spaces

Joseph A. Ball·Gregory Marx·Victor Vinnikov
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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.

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Joseph A. Ball, Gregory Marx, Victor Vinnikov. 2016-02-02. Noncommutative reproducing kernel Hilbert spaces. https://arxiv.org/abs/1602.00760

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