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.

AMS Subject Classification (1991): 47D03, 47A10, 46J25

Keyword(s): Favard class, one-parameter semigroup, pseudo-resolvent, uniformly Lipschitz vector

Received January 16, 1998. (Registered under 3316/2009.)