On the generators of Clifford semigroups: polynomial resolvents and their integral transforms

TL;DR

This paper explores the invertibility and integral representations of polynomial functions of generators of Clifford semigroups in Banach spaces, extending classical resolvent concepts to Clifford algebra contexts.

Contribution

It introduces new integral formulas for polynomial resolvents and the $S$-resolvent operator in Clifford algebra settings, generalizing existing operator theory.

Findings

Derived integral representation for $P( extsf{A})^{-1}$ using Laplace transforms.

Provided a new integral formula for the operator $( extsf{A}^2 - 2 ext{Re}(q) extsf{A} + |q|^2)^{-1}$.

Reproved the integral representation of the $S$-resolvent operator for $ extsf{A}$.

Abstract

This paper deals with generators $\mathsf{A}$ of strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra $\mathit{C}\ell(0,n)$. We study the invertibility of operators of the form $P(\mathsf{A})$, where $P(x)\in\mathbb{R}[x]$ is any real polynomial, and we give an integral representation for $P(\mathsf{A})^{-1}$ by means of a Laplace-type transform of the semigroup $\mathsf{T}(t)$ generated by $\mathsf{A}$. In particular, we deduce a new integral representation for the operator $(\mathsf{A}^2 - 2\mathrm{Re}(q) \,\mathsf{A} + |q|^2)^{-1}$. As an immediate consequence, we also obtain a new proof of the well-known integral representation for the $S$-resolvent operator of $\mathsf{A}$ (also called spherical resolvent operator of $\mathsf{A}$).