Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

TL;DR

This paper develops a unified spectral theory for Clifford operators using fine structures on the $S$-spectrum, connecting polyharmonic, polyanalytic, and monogenic functions through integral representations and functional calculi.

Contribution

It introduces the concept of fine structures on the $S$-spectrum within Clifford analysis, linking different classes of functions and their associated calculi.

Findings

Monogenic and $F$-functional calculi coincide for Clifford operators.

The paper extends spectral theory to polyharmonic and polyanalytic functions.

Integral representation formulas unify various functional calculi in Clifford analysis.

Abstract

The spectral theory on the $S$-spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the $S$-spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the $F$-functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.