LOCAL C-SEMIGROUPS AND LOCAL C-COSINE FUNCTIONS

This paper is concerned with the local C-semigroups and local C-cosine functions without the assumption that the image of C is dense in a Banach space X. First, the sufficient and necessary conditions for a local C-semigroup and a C-cosine function to be the restriction of a global C-semigroup and a global C-cosine function to an interval are given, respectively. Secondly, it is characterized for a closed operator to be the generator of a local C-semigroup and a local C-cosine function, respectively.