An introduction to hyperholomorphic spectral theories and fractional powers of vector operators

TL;DR

This paper surveys spectral theories related to hyperholomorphic functions in higher dimensions, focusing on quaternionic and paravector variables, and explores their applications in defining fractional Fourier's law for nonhomogeneous materials.

Contribution

It introduces and compares two hyperholomorphic spectral theories based on slice hyperholomorphic and monogenic functions, highlighting their differences, relations, and applications.

Findings

The $S$-spectrum-based theory applies to fractional Fourier's law.

The monogenic spectrum provides an alternative spectral framework.

Relations between the two theories are established via the $F$-functional calculus.

Abstract

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for $n$-tuples of operators $(A_1,...,A_n)$. A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the $S$-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the $F$-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. Here we also discuss how to define the fractional Fourier's law for nonhomogeneous materials, such definition is based on the spectral theory on the $S$-spectrum.