Pseudoresolvents and Semigroups

Abstract

The notions of resolvent, pseudoresolvent and a few results along with some remarkable properties are recalled. A new concept, the L1-type pseudoresolvent is introduced. The aim of this work is firstly to give a characterization theorem for L1-type pseudoresolvents and for the generators of L1-type pseudoresolvents. Moreover, the connection between the L1-type pseudoresolvents and C0-equicontinuous semigroups is pointed out. Secondly, the main part of this work is devoted to approximation of pseudoresolvents and their generators. If Rn; R : ! L(X); n 1 are generated pseudoresolvents and An; A their generators, then it is investigated under which conditions A is approximated by An and R is approximated by Rn; n 1: In addition, the conditions under which a sequence of generated pseudoresolvents approximates a pseudoresolvent are given, and in this case the connection between generators is studied. In the last chapter we have proved a theorem of characterization for exponentially bounded semigroups. To any exponentially bounded semigroup we have associated a projective family of semigroups acting on Banach spaces.