Operator Algebra Generated by an Element from the Module B(V1, V2)

Abstract

Let B∈ B(V1) , A∈ B(V2) and X∈ B(V1, V2) be given operators. In this paper the algebra AAXB generated by AXB is defined. Some results are obtained regarding invertibility in AAXB. Special cases and extensions are discussed using algebraic representations. Applications to the operator equations AX= C, XB= C, AXB= C, X- AXB= C and AX- XB= C are illustrated.