A remark on generalised Putnam-Fuglede theorems
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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.References
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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
- Received by editor(s): September 30, 1998
- Published electronically: September 14, 2000
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 83-87
- MSC (1991): Primary 47B20, 47B15
- DOI: https://doi.org/10.1090/S0002-9939-00-05920-7
- MathSciNet review: 1784016