A new functional calculus for non-commuting operators

TL;DR

This paper introduces a novel functional calculus for non-commuting operator tuples using slice monogenic functions, enabling explicit eigenvalue equations and a new spectrum concept, extending classical calculus.

Contribution

It presents a new functional calculus based on slice monogenic functions for non-commuting operators, distinct from previous approaches, with explicit eigenvalue construction.

Findings

Defines a new spectrum for operator tuples

Constructs explicit eigenvalue equations

Extends Riesz-Dunford calculus to non-commuting cases

Abstract

In this paper we use the notion of slice monogenic functions \cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.