An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

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 $$(A_1,\ldots ,A_n)$$ ( 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. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

Riesz projection and essential $S$-spectrum in quaternionic setting

“…Over the recent years, the spectral theory for quaternionic operators has piqued the interest and attracted the attention of multiple researchers, see for instance [1,4,12,13,14,15,16,17,29] and references therein. Research in this topic is motivated by application in various fields, including quantum mechanics, fractional evolution problems [14], and quaternionic Schur analysis [3].…”