Noncommutative reproducing kernel Hilbert spaces

TL;DR

This paper develops a theory of positive kernels and reproducing kernel Hilbert spaces tailored for free noncommutative functions, expanding tools for noncommutative analysis and its applications.

Contribution

It introduces a framework for positive kernels and RKHS in free noncommutative function theory, bridging a gap in noncommutative analysis tools.

Findings

Established a theory of positive kernels for noncommutative functions

Constructed associated reproducing kernel Hilbert spaces in the noncommutative setting

Provided foundational results for applications in noncommutative analysis

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.