Formulations of the F-functional calculus and some consequences

TL;DR

This paper introduces two formulations of the F-functional calculus based on the Fueter-Sce mapping theorem, explores the F-resolvent equation, and studies quaternionic analogues, advancing the theoretical framework of functional calculus in hypercomplex analysis.

Contribution

It presents two new formulations of the F-functional calculus, establishes the F-resolvent equation in dimension 3, and investigates quaternionic extensions, enriching the mathematical theory.

Findings

Proved the F-resolvent equation in dimension 3

Introduced the pseudo F-resolvent equation

Studied quaternionic version of the F-functional calculus

Abstract

In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove the F-resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the F-functional calculus.