On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenCⁿ(A, B) X=0, n some natural number, implies thatC (A, B)X=C(A^*,B^*)X=0. Secondly, it is shown that ifA andB^* are contractions withC₀ completely non-unitary parts, thenRⁿ(A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A^*,B^*)X=C (A, B^*)X=C (A^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.