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

Abstract

The aim of this article is to introduce the $H^{\infty }$-functional calculus for unbounded bisectorial operators in a Clifford module over the algebra ${\mathbb{R}}_n$. This work is based on the universality property of the $S$-functional calculus, which shows its applicability to fully Clifford operators. 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 $H^{\infty }$- 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.