Approximation of Pseudoresolvents

Abstract.

After a short introduction where a few main notions and results are recalled, we state some results connected with generated and L_∞-type pseudoresolvents. Among others, we give theorems of characterization both for L_∞-type pseudoresolvents and for their generators. The relation between L_∞-type pseudoresolvents and C₀-equicontinuous semigroups is also pointed out. The last part of the present paper is devoted to approximation of pseudoresolvents and their generators. If $R_{n}, R : \Lambda \rightarrow L(X), n \geq 1$ are generated pseudoresolvents and A _n , A their generators, we investigate conditions under which A is approximated by A _n and R is approximated by R _n , n ≥ 1. In addition, we give conditions under which a sequence of generated pseudoresolvents approximates a pseudoresolvent, and in this case we study the connection between generators.