An introduction to the fine structures on the $S$-spectrum

TL;DR

This paper reviews the second step of the Fueter-Sce mapping theorem, exploring how it transforms slice hyperholomorphic functions into axially monogenic functions, thereby enriching the spectral theories on the $S$-spectrum.

Contribution

It provides a comprehensive analysis of the operator that maps slice hyperholomorphic functions to axially monogenic functions, revealing multiple factorizations and associated functional calculi.

Findings

Multiple factorizations of the Fueter-Sce operator are identified.

Various function spaces and calculi are generated through these factorizations.

The work enhances understanding of the fine structures in spectral theories on the $S$-spectrum.

Abstract

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.