The S-functional calculus for the Clifford adjoint operator

TL;DR

This paper develops the S-functional calculus for Clifford operators, showing the spectrum's invariance under adjoint and establishing a link between left and right calculi for T and T* in the Clifford algebra framework.

Contribution

It introduces a comprehensive S-functional calculus for Clifford operators, demonstrating spectrum invariance under adjoint and connecting left and right calculi via a specific function.

Findings

The S-spectrum of T* equals that of T.

Bisectoriality of T transfers to T*.

Explicit link between left calculus of T and right calculus of T*.

Abstract

In this paper, we work within the framework of modules over the Clifford algebra $\mathbb{R}_d$. Our investigation focuses on the S-spectrum and the S-functional calculus in its various forms for the adjoint T* of a Clifford operator T. One of the key results we present is that the bisectoriality of T can be transferred to T*. This is grounded in the fact that, for Clifford operators, the S-spectrum of the adjoint operator T* is identical to that of T. Furthermore, we demonstrate that for the existing formulations of the S-functional calculus, including bounded, unbounded, $\omega$, and $H^\infty$ versions, there is a clear connection between the left functional calculus of T and the right functional calculus of T*. This explicit link between the left functional calculus of T and the right functional calculus of T* and vice versa is obtained using the function $f^\#(s):=\overline{f(\overline{s})}$. Finally, we discuss the fact that the S-spectrum, related to the invertibility of the second-order operator T^2-2s_0T+|s|^2, can actually be characterized through the invertibility of the first-order $\mathbb{R}$-linear operator T-I^Rs.