An operator version of the Korovkin theorem revisited

Abstract

Let T be a compact Hausdorff space. We prove that if A:C([a,b])⟶C(T) is a positive linear operator such that A(1)A(e2)=[A(e1)]2 and A(1)(t)>0 for all t∈T then A is exactly the weighted composition operator defined by A(f)=A(1)f∘(A(e1)A(1)) for every f∈C([a,b]).