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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A remark on generalised Putnam-Fuglede theorems


Author: B. P. Duggal
Journal: Proc. Amer. Math. Soc. 129 (2001), 83-87
MSC (1991): Primary 47B20, 47B15
Published electronically: September 14, 2000
MathSciNet review: 1784016
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Abstract:

Given $A, B \in B (H)$, the algebra of operators on a Hilbert space $H$, define $\delta_{A, B} : B(H) \rightarrow B(H)$ and $\triangle_{A, B} : B(H) \rightarrow B(H)$ by $\delta_{A, B}(X) = AX-XB$ and $\triangle_{A, B}(X) = AXB - X$. Let $P_1$ and $P_2$ be two classes of operators strictly larger than the class of normal operators. Define $(P_1, P_2) \in PF (\delta)$ (resp., $PF (\triangle))$ if $ker \delta _{A,B} \subset ker \delta_{A^*, B^*}$ (resp., $ker \triangle_{A, B} \subset ker \triangle_{A^*, B^*})$ for all $A \in P_1$and $B^* \in P_2$. This note shows that the equivalence $(P_1, P_2) \in PF (\delta) \Longleftrightarrow (P_1, P_2) \in PF(\triangle)$ holds for a number of the commonly considered classes of operators.


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Additional Information

B. P. Duggal
Affiliation: Department of Mathematics, Faculty of Science, University of Botswana, P/Bag 0022, Gaborone, Botswana, Southern Africa
Address at time of publication: Department of Mathematics, Faculty of Science, United Arab Emirates University, P.O. Box 17551, Al Ain, Arab Emirates
Email: duggbp@mopipi.ub.bw, bpduggal@uaeu.ac.ae

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05920-7
PII: S 0002-9939(00)05920-7
Keywords: Putnam-Fuglede theorem, subnormal/p-hyponormal/dominant operators
Received by editor(s): September 30, 1998
Published electronically: September 14, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society