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.