GENERATION OF LOCAL $C$-SEMIGROUPS AND SOLVABILITY OF THE ABSTRACT CAUCHY PROBLEMS

Abstract

For a bounded linear injection $C$ on a Banach space $X$ and a (not necessarily densely defined) closed linear operator $A$ which commutes with $C$, we present various conditions for $A$ to generate a local $C$-semigroup. A Hille-Yosida type generation theorem in terms of the asymptotic $C$-resolvent of $A$ is proved, and various characterizations of a generator by means of existence of unique strong solutions of the associated abstract Cauchy problems are obtained.