On the Favard classes of semigroups associated with pseudo-resolvents

Abstract

A pseudo-resolvent on a Banach space, indexed by positive numbers and tempered at infinity, gives rise to a bounded strongly continuous one-parameter semigroup S on a closed subspace of the ambient Banach space. We prove that the range space of the pseudo-resolvent contains the domain of the generator of S, and is contained in the Favard class of S, which consists of all uniformly Lipschitz vectors for S. We explore when some or all of these three spaces coincide.