On the generalized Sylvester operator equation AX−Y B = C

Abstract

In this paper, we study the necessary and sufficient conditions to have a solution pair ( X , Y ) to the generalized Sylvester equation AX−Y B = C where A , B , C ∈ L ( H ) are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair ( X , Y ) of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators.