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

This paper deals with generators Aof strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra Cl(0, n). We study the invertibility of operators of the form P(A), where P(x) ∈ R[x] is any real polynomial, and we give an integral representation for P(A)−1 by means of a Laplace-type transform of the semigroup T(t) generated by A. In particular, we deduce a new integral representation for the spherical quadratic resolvent of A (also called pseudoresolvent of A). As an immediate consequence, we also obtain a new proof of the well-known integral representation for the spherical resolvent of A.