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

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 $P(\mathrm{x})\in\mathbb{R}[\mathrm{x}]$ 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 spherical quadratic resolvent of $\mathsf{A}$ (also called pseudoresolvent of $\mathsf{A}$). As an immediate consequence, we also obtain a new proof of the well-known integral representation for the spherical resolvent of $\mathsf{A}$.