The resolvent equations for the Harmonic and bi-Harmonic functional calculi in dimension five

TL;DR

This paper explores the structure of resolvent equations for harmonic and bi-harmonic functional calculi based on the $S$-spectrum in five dimensions, revealing new properties and differences from classical cases.

Contribution

It introduces and analyzes the resolvent equations for harmonic and bi-harmonic functional calculi in dimension five, highlighting their unique features and differences from classical resolvent equations.

Findings

Derived the harmonic and bi-harmonic resolvent equations in dimension five.

Established product rules and Riesz projectors for these calculi.

Showed significant differences from classical Cauchy kernel-based resolvent equations.

Abstract

The fine structures on the $S$-spectrum constitute a new research area that includes a class of functional calculi based on the $S$-spectrum and on integral transforms determined by the Fueter--Sce mapping theorem and the Cauchy formula for slice hyperholomorphic functions. This strategy, based on integral transforms, allows us to construct functional calculi that include harmonic and polyharmonic functional calculi. The resolvent operators in this setting do not arise directly from a Cauchy kernel, but rather from suitable manipulations of it. For this reason the corresponding resolvent equations differ substantially from those associated with the classical Cauchy kernel. In this paper, we investigate the harmonic and biharmonic resolvent equations in dimension five, as well as the corresponding product rules and Riesz projectors for these functional calculi.