Noncommutative reproducing kernel Hilbert spaces

By Joseph A. Ball, Gregory Marx, Victor Vinnikov. 2016

66 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/220c04b07bbe1901676b821c065df4f984a0cae2

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “We now define the nc Drury–Arveson space, which was introduced by Ball, Marx and Vinnikov in [14]; this nc reproducing Hilbert space will play a central role in this paper.”
Global Holomorphic Functions in Several Noncommuting Variables (2015, methodology): “For finite-dimensional K and L, this was done in [2]; for infinite-dimensional K and L the formula was proved in [6], using results from [7].”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “en questions concerning full noncommutative envelopes and noncommutative-Zariski closure of any ﬁnite subset Ω of a full noncommutative set Ξ, as well as a review of material from our companion paper [23] concerning completely positive noncommutative kernels, a notion needed for the very formulation”