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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 normal derivations


Author: B. P. Duggal
Journal: Proc. Amer. Math. Soc. 126 (1998), 2047-2052
MSC (1991): Primary 47A30, 47A63, 47B15, 47B48
MathSciNet review: 1451795
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Abstract: Given a Hilbert space $H$, let $A,S$ be operators on $H$. Anderson has proved that if $A$ is normal and $AS=SA$, then $\|AX-XA+S\|\ge\|S\|$ for all operators $X$. Using this inequality, Du Hong-Ke has recently shown that if (instead) $ASA=S$, then $\|AXA-X+S\|\ge\|A\|^{-2}\|S\|$ for all operators $X$. In this note we improve the Du Hong-Ke inequality to $\|AXA-X+S\|\ge\|S\|$ for all operators $X$. Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.


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Additional Information

B. P. Duggal
Affiliation: Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman
Email: duggbp@squ.edu.om

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04326-3
PII: S 0002-9939(98)04326-3
Keywords: Normal derivation, norm inequality, contraction, unitarily invariant norm.
Received by editor(s): January 31, 1996
Received by editor(s) in revised form: December 18, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society