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

 

Commutator inequalities associated with the polar decomposition


Author: Fuad Kittaneh
Journal: Proc. Amer. Math. Soc. 130 (2002), 1279-1283
MSC (2000): Primary 15A23, 15A57, 15A60, 47A30, 47B47
Published electronically: October 12, 2001
MathSciNet review: 1879948
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Abstract: Let $A=UP$ be a polar decomposition of an $n\times n$ complex matrix $A$. Then for every unitarily invariant norm $\vert\vert\vert\cdot\vert\vert\vert$, it is shown that

\begin{displaymath}\vert\vert\vert\, \vert UP-PU\vert^2\vert\vert\vert \le \vert... ...vert\le \Vert UP+PU\Vert\,\vert\vert\vert UP-PU\vert\vert\vert,\end{displaymath}

where $\Vert\cdot\Vert$ denotes the operator norm. This is a quantitative version of the well-known result that $A$ is normal if and only if $UP=PU$. Related inequalities involving self-commutators are also obtained.


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Additional Information

Fuad Kittaneh
Affiliation: Department of Mathematics, University of Jordan, Amman, Jordan
Email: fkitt@ju.edu.jo

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06197-4
PII: S 0002-9939(01)06197-4
Keywords: Commutator, polar decomposition, positive semidefinite matrix, unitarily invariant norm
Received by editor(s): December 14, 1999
Received by editor(s) in revised form: November 1, 2000
Published electronically: October 12, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society