An operator version of the Korovkin theorem

Abstract

We prove the following operator version of the Korovkin theorem: Let T be a compact Hausdorff space, Vn:C[a,b]→C(T) a sequence of linear positive operators and A:C[a,b]→C(T) a linear positive operator such that A(1)A(e2)=[A(e1)]2 and A(1)(t)>0, ∀t∈T. If limn→∞⁡Vn(1)=A(1), limn→∞⁡Vn(e1)=A(e1), limn→∞⁡Vn(e2)=A(e2) all uniformly on T, then for every f∈C[a,b], limn→∞⁡Vn(f)=A(f) uniformly on T. Many and various examples are given.