Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-01-06197-4
Published electronically: October 12, 2001
MathSciNet review: 1879948
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 15A23, 15A57, 15A60, 47A30, 47B47

Retrieve articles in all journals with MSC (2000): 15A23, 15A57, 15A60, 47A30, 47B47


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

American Mathematical Society