On some factorizations of operators

Given two subsets A and B of the algebra of bounded linear operators on a Hilbert space H we denote by AB:={AB:A∈A,B∈B}. The goal of this article is to describe AB if A and B denote classes of projections, partial isometries, positive (semidefinite) operators, etc. Moreover, fixed T∈AB we shall describe (AB)T:={(A,B)∈A×B:AB=T}, p1((AB)T):={A∈A:T=ABfor someB∈B} and p2((AB)T):={B∈B:T=ABfor someA∈A}.