Exponential stability and unstability of semigroups of linear operators in Banach spaces

Abstract

Necessary and sufficient conditions for uniform exponential stability and uniform exponential unstability of semigroups of linear operators are given, in terms of Banach function spaces and Banach sequence spaces, respectively. We generalize some well-known theorems proved by Datko, Pazy, Rolewicz and Neerven concerning the exponential stability of C0 semigroups. We shall obtain the versions of the theorems mentioned above, for the case of uniform exponential unstability of C0 -semigroups. Mathematics subject classification (2000): 34D05, 34D20, 47D06.