Approximation of *Weak-to-Norm Continuous Mappings

The purpose of this paper is to study the approximation of vector-valued mappings defined on a subset of a normed space. We investigate Korovkin-type conditions useful to recognize if a given sequence of linear operators is a so-called approximation process. First, we give a sufficient condition for this sequence to approximate the class of bounded, uniformly continuous functions. Then we present some sufficient and necessary conditions guaranteeing the approximation within the class of unbounded, *weak-to-norm continuous mappings. We also derive some estimates of the rate of convergence. We apply concrete approximation processes to derive representation formulae for semigroups of bounded linear operators.