Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions

TL;DR

This paper develops an analytic functional calculus for Clifford operators using complex spectrum and stem functions, introducing a new approach that differs from previous methods based on slice regular functions.

Contribution

It constructs a novel functional calculus for Clifford operators utilizing complex spectra and stem functions, expanding the theoretical framework beyond prior slice regular function methods.

Findings

Established a spectrum defined in the complex plane for Clifford operators

Proved the existence of a Cauchy type transform inducing an isomorphism

Developed an analytic functional calculus based on stem functions

Abstract

The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.