Non-commutative Functional Calculus and Spectral Theory

TL;DR

This paper introduces a new functional calculus for non-commuting elements in Banach algebras using free analytic functions, expanding spectral theory in non-commutative settings.

Contribution

It develops a novel functional calculus framework for non-commuting operators based on free analytic functions, advancing non-commutative spectral theory.

Findings

Established a functional calculus for non-commuting elements

Extended spectral theory to non-commutative Banach algebra elements

Utilized free analytic functions bounded on polynomial polyhedra

Abstract

We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.