The $H^\infty$-functional calculus for bisectorial Clifford operators

TL;DR

This paper develops an H-infinity functional calculus for unbounded bisectorial operators within Clifford modules, extending previous work to include functions with polynomial growth and regularizable functions, broadening the scope of operator analysis.

Contribution

It introduces the H-infinity functional calculus for unbounded Clifford operators, including polynomially growing functions, and establishes a regularization method for defining this calculus.

Findings

Extended functional calculus to unbounded Clifford operators.

Included polynomial growth functions in the calculus.

Provided a regularization procedure for broader function classes.

Abstract

The aim of this article is to introduce the H-infinity functional calculus for unbounded bisectorial operators in a Clifford module over the algebra R_n. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the omega-functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H-infinity functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.