Convergence theorems and Tauberian theorems for functions and sequences in Banach spaces and Banach lattices

摘要

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order gamma - 1 with gamma > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesaro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators.