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

DOI:
https://doi.org/10.1090/S0002-9939-01-06197-4

Published electronically:
October 12, 2001

MathSciNet review:
1879948

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polar decomposition of an complex matrix . Then for every unitarily invariant norm , it is shown that

where denotes the operator norm. This is a quantitative version of the well-known result that is normal if and only if . Related inequalities involving self-commutators are also obtained.

**1.**R. Bhatia,*Matrix Analysis*, Springer-Verlag, New York, 1997. MR**98i:15003****2.**R. Bhatia and F. Kittaneh,*On some perturbation inequalities for operators*, Linear Algebra Appl.**106**(1988), 271-279. MR**90f:47007****3.**R. Bhatia and F. Kittaneh,*On the singular values of a product of operators*, SIAM J. Matrix Anal. Appl.**11**(1990), 272-277. MR**90m:47033****4.**R. Bhatia and F. Kittaneh,*Notes on matrix arithmetic-geometric mean inequalities*, Linear Algebra Appl.**308**(2000), 203-211. MR**2001a:15023****5.**C. K. Fong,*Norm estimates related to self-commutators*, Linear Algebra Appl.**74**(1986), 151-156. MR**87c:15047****6.**P. R. Halmos,*Finite-Dimensional Vector Spaces*, Springer-Verlag, New York, 1974. MR**53:13258****7.**O. Hirzallah and F. Kittaneh,*Matrix Young inequalities for the Hilbert-Schmidt norm*, Linear Algebra Appl.**308**(2000), 77-84. MR**2001b:15027****8.**R. A. Horn and C. R. Johnson,*Matrix Analysis*, Cambridge University Press, Cambridge, 1985. MR**87e:15001****9.**F. Kittaneh,*A note on the arithmetic-geometric mean inequality for matrices*, Linear Algebra Appl.**171**(1992), 1-8. MR**93c:15018****10.**F. Kittaneh,*On some operator inequalities*, Linear Algebra Appl.**208/209**(1994), 19-28. MR**95d:47015****11.**F. Kittaneh,*Norm inequalities for certain operator sums*, J. Funct. Anal.**143**(1997), 337-348. MR**97k:47005****12.**F. Kittaneh,*Norm inequalities for sums of positive operators*, J. Operator Theory, to appear.

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Additional Information

**Fuad Kittaneh**

Affiliation:
Department of Mathematics, University of Jordan, Amman, Jordan

Email:
fkitt@ju.edu.jo

DOI:
https://doi.org/10.1090/S0002-9939-01-06197-4

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