We present some results on
collectively compact operator approximation theory in locally convex Hausdorff
spaces (l.c.s.). The notion of a collectively compact family of operators acting on a
Banach space has been introduced by Anselone and Palmer in connection with the
numerical solution of integral equations. Meanwhile collectively compact families of
operators have been studied in general topological vector spaces. In contrast to
those investigations dedicated to the characterization of collectively compact
families of operators the present paper focuses on spectral approximation
theorems in l.c.s. similar to those given by Anselone and Palmer in the case of
Banach spaces. In doing this it turns out that the notion of the spectrum,
which causes no problems in Banach algebra theory, entails some difficulty. A
way out is indicated by using notions and tools of locally convex algebra
theory.
