Abstract
LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BeB(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*eB (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.
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