Global holomorphic functions in several non-commuting variables
Jim Agler, John E. McCarthy
TL;DR
This paper introduces the concept of free holomorphic functions in several non-commuting variables, establishing their approximation by free polynomials and providing key theorems like realization and Oka-Weil.
Contribution
It defines free holomorphic functions and proves their approximation by free polynomials, along with realization and Oka-Weil theorems for free analytic functions.
Findings
Free holomorphic functions are locally uniformly approximable by free polynomials.
Established a realization formula for free analytic functions.
Proved an Oka-Weil theorem for free holomorphic functions.
Abstract
We define a free holomorphic function to be a function that is locally a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.
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Taxonomy
TopicsHolomorphic and Operator Theory · Algebraic and Geometric Analysis · Meromorphic and Entire Functions