Annals of Functional Analysis

Singular value and‎ ‎arithmetic-geometric mean inequalities for operators

Hussien Albadawi

Full-text: Open access


‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

Article information

Ann. Funct. Anal. Volume 3, Number 1 (2012), 10-18.

First available in Project Euclid: 12 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 15A18‎ ‎47A63‎ ‎47B10

Singular value ‎unitarily invariant norm ‎positive‎ ‎operator ‎arithmetic-geometric mean inequality


Albadawi, Hussien. Singular value and‎ ‎arithmetic-geometric mean inequalities for operators. Ann. Funct. Anal. 3 (2012), no. 1, 10--18. doi:10.15352/afa/1399900020.

Export citation


  • H. Albadawi, Hölder-type inequalities involving unitarily invariant norms, Positivity (to appear).
  • T. Ando, Majorizations and inequalities in matrix theory, Linear Algebra Appl. 199 (1994), 17–67.
  • T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.
  • K. Audenaert, A singular value inequality for Heinz means, Linear Algebra Appl. 422 (2007), 279–283.
  • R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.
  • R. Bhatia and C. Davis, More matrix forms of the arithmetic–geometric mean inequality, SIAM J. Matrix Anal. Appl. 14 (1993), 132–136.
  • R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11 (1990), 272–277.
  • R. Bhatia and F. Kittaneh, Notes on matrix arithmetic–geometric mean inequalities, Linear Algebra Appl. 308 (2000), 203–211.
  • O. Hirzallah, Inequalities for sums and products of operators, Linear Algebra Appl. 407 (2005), 32–42.
  • R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
  • F. Kittaneh, A note on the arithmetic–geometric mean inequality for matrices, Linear Algebra Appl. 171 (1992), 1–8.
  • H. Kosaki, Arithmetic–geometric mean and related inequalities for operators, J. Funct. Anal. 156 (1998), 429–451.
  • R. Mathias, An arithmetic–geometric–harmonic mean inequality involving Hadamard products, Linear Algebra Appl. 184 (1993), 71–78.
  • K. Shebrawi and H. Albadawi, Norm inequalities for the absolute value of Hilbert space operators, Linear and Multilinear Algebra, 58 (2010), no. 4, 453–463.
  • X. Zhan, Singular values of differences of positive semidefinite matrices, SIAM J. Matrix. Anal. Appl. 22 (2000), 819–823.
  • X. Zhan, Matrix Inequalities, Springer-Verlag, Berlin, 2002.