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.
