Functions of the infinitesimal generator of a strongly continuous quaternionic group

TL;DR

This paper extends the functional calculus for quaternionic operators by defining bounded operators using the quaternionic Laplace-Stieltjes transform, broadening the class of functions applicable beyond slice regular functions.

Contribution

It introduces a new approach to define functions of quaternionic generators using Laplace-Stieltjes transform, connecting it with existing calculus and addressing invertibility.

Findings

Defined bounded operators f(T) for a larger class of functions

Connected the new calculus with existing quaternionic functional calculus

Studied the invertibility of the operators f(T)

Abstract

The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t \in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S$-spectrum of $T$ but not necessarily at infinity. Moreover, we establish the relation of $f(T)$ with the quaternionic functional calculus and we study the problem of finding the inverse of $f(T)$.