PSEUDORESOLVENTS AND SEMIGROUPS

Abstract

The notions of resolvent, pseudoresolvent and a few results along with some re- markable properties are recalled. A new concept, the L ∞ -type pseudoresolvent is introduced. The aim of this work is firstly to give a characterization theorem for L ∞ -type pseudoresolvents and for the generators of L ∞ -type pseudoresolvents. Moreover, the connection between the L ∞ -type pseudoresolvents and C 0 -equicontinuous semigroups is pointed out. Secondly, the main part of this work is devoted to approximation of pseudoresol- vents and their generators. If R n , R : Λ → L(X), n ≥ 1 are generated pseudoresol- vents and A n , A their generators, then it is investigated under which conditions A is approximated by A n and R is approximated by R n , 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.