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A Hilbert-Schmidt norm inequality associated with the Fuglede-Putnam theorem

Published online by Cambridge University Press:  17 April 2009

Takayuki Furuta
Takayuki Furuta
Affiliation:
Department of Mathematics, Faculty of Science, Hirosaki University, Bunkyo–Cho 3, 036 Aomori, Japan.
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Abstract

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The familiar Fuglede-Putnam theorem asserts that AX = XB implies A*X = XB* when A and B are normal. We prove that A and B* be hyponormal operators and let C be a hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert-Schmidt operator X, the Hilbert-Schmidt norm of AXD + CXB is greater than or equal to the Hilbert-Schmidt norm of A*XD* + C*XB*. In particular, AXD = CXB implies A*XD* = C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert-Schmidt operator of the hypothesis on X to be every operator and still retain the inequality under some suitable hypotheses.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 2 , April 1982 , pp. 177 - 185
Copyright
Copyright © Australian Mathematical Society 1982

References

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