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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
DOI: https://doi.org/10.1090/S0002-9939-00-05920-7
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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  • [1] S.K. Berberian, Approximate proper vectors, Proc., Amer. Math. Soc. 13(1962), 111-114. MR 24:A3516
  • [2] S.K. Berberian, Extensions of a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 71(1978), 113-114. MR 58:7176
  • [3] R.G. Douglas, On the operator equation $S^*XT = X $ and related topics, Acta Sci. Math. (Szeged) 30(1969), 19-32. MR 40:3347
  • [4] B.P. Duggal, On intertwining operators, Mh. Math. 106(1988), 139-148. MR 89k:47031
  • [5] B.P. Duggal, On generalised Putnam-Fuglede theorems, Mh. Math. 107(1989), 309-332. MR 90g:47045
  • [6] B.P. Duggal, On quasi-similar p-hyponormal operators, Integ. Equat. Oper. Th. 26(1996), 338-345. MR 98g:47019
  • [7] Ming Fan, An asymmetric Putnam-Fuglede theorem for $\Theta$-class operators and some related topics (Chinese), J. Fudan Univ. Natur. Sci. 26(1987), 347-350. MR 89f:47034
  • [8] P.R. Halmos, A Hilbert Space Problem Book, Springer-Verlag (1982). MR 84e:47001
  • [9] Jin Chuan Hou, On Putnam-Fuglede theorems for non-normal operators (Chinese), Acta Math. Sinica 28(1985), 333-340. MR 87b:47022
  • [10] M. Radjabalipour, On majorization and normality of operators, Proc. Amer. Math. Soc. 62(1977), 105-110. MR 55:3856
  • [11] M. Radjabalipour, An extension of Putnam-Fuglede theorem for hyponormal operators, Math. Z. 194(1987), 117-120. MR 88c:47065
  • [12] H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34(1971), 653-665. MR 43:3829
  • [13] J.G. Stampfli and B.L. Wadhwa, An asymmetric Putnam-Fugledfe theorem for dominant operators, Indiana Univ. Math. J. 25(1976), 359-365. MR 53:14197
  • [14] Xia Ming Wang, On operator valued roots of commutative analytic functions (Chinese), Chinese Ann. Math. Ser. A 12(1991), 65-69. MR 92e:47033
  • [15] Tong Yusun, Kernels of generalized derivations, Acta Sci. Math. (Szeged), 54(1990), 159-169. MR 92f:47032

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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: https://doi.org/10.1090/S0002-9939-00-05920-7
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

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