The $H^\infty$-functional calculus for right slice hyperholomorphic functions and right linear Clifford operators

TL;DR

This paper extends the $H^ Infty$-functional calculus to right slice hyperholomorphic functions and right linear Clifford operators, resolving a key issue caused by non-commutative multiplication.

Contribution

It introduces a method to define the $H^ Infty$-functional calculus for right slice hyperholomorphic functions on right linear Clifford operators, overcoming previous limitations.

Findings

Established the $H^ Infty$-functional calculus for right slice hyperholomorphic functions.

Resolved the issue of non-commutative multiplication in the calculus extension.

Demonstrated applicability to sectorial and bisectorial operators.

Abstract

In 2016, the spectral theory on the $S$-spectrum was used to establish the $H^\infty$-functional calculus for quaternionic or Clifford operators. This calculus applies for example to sectorial or bisectorial right linear operators $T$ and left slice hyperholomorphic functions $f$ that can grow as polynomials. It relies on the product of the two operators $e(T)^{-1}$ and $(ef)(T)$, both defined via some underlying $S$-functional calculus (also called $\omega$-functional calculus). For left slice holomorphic functions $f$ this definition does not depend on the choice of the regularizer function $e$. However, due to the non-commutative multiplication of Clifford numbers, it was unclear how to extend this definition to right slice hyperholomorphic functions. This paper addresses this significant unresolved issue and shows how right linear operators can possess the $H^\infty$-functional calculus also for right slice hyperholomorphic functions.