On intertwining operators |
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Authors: | B P Duggal |
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Institution: | (1) School of Mathematical Sciences, University of Khartoum, P.O. Box 321, Khartoum, Sudan |
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Abstract: | LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,B B(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, thenC
n
(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 withC
0 completely non-unitary parts, thenR
n
(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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Keywords: | |
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