## Abstract

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.

Show Summary Details# Extensions of Three Matrix Inequalities to Semisimple Lie Groups

#### Open Access

## Abstract

## References

## About the article

More options …# Special Matrices

More options …

Editor-in-Chief: da Fonseca, Carlos Martins

Covered by:

WoS (ESCI)

SCOPUS

MathSciNet

zbMATH

CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.408

Source Normalized Impact per Paper (SNIP) 2018: 1.005

Mathematical Citation Quotient (MCQ) 2018: 0.29

Xuhua Liu / Tin-Yau Tam

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.

Keywords: Araki-Lieb-Thirring inequality; positive definite matrices; semisimple Lie groups; log majorization; Kostant’s pre-order

[1] Araki, H., On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), 167–170. Google Scholar

[2] Audenaert, K. M. R., On the Araki-Lieb-Thirring Inequality, Int. J. Inf. Syst. Sci. 4 (2008), 78–83. Google Scholar

[3] Bhatia, R., “Matrix Analysis", Springer-Verlag, New Yor, 1997. Google Scholar

[4] Helgason, S., “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, 1978. Google Scholar

[5] Hiai, F., Log-majorizations and norm inequalities for exponential operators, Linear Operators, Volume 38, p.119–181, Banach Center Publ., Polish Acad. Sci., Warsaw, 1997. Google Scholar

[6] Horn, A., Doubly stochastic matrices and the diagonal of a rotation of matrix, Amer. J. Math. 76 (1954), 620–630. Google Scholar

[7] Knapp, A. W., “Lie Groups beyond an Introduction", 2nd ed., Birkhäuser, 2002. Google Scholar

[8] Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455. Google Scholar

[9] Lieb, E., and Thirring,W., in Studies inMathematical Physics (Eds. E. Lieb, B. Simon and A.Wightman), p.301–302, Princeton Press, 1976. Google Scholar

[10] Marshall, A. W., I. Olkin, and B. C. Arnold, “Inequalities: Theory of Majorization and its Applications (2nd ed.)”, Springer, 2011. Google Scholar

[11] Simon, B., “Trace Ideals and Their Applications”, London Mathematical Society Lecture Note Series, 35, Cambridge Univ. Press, 1979. Google Scholar

[12] Tam, T.Y., Kostant’s convexity theorem and the compact classical groups, Linear and Multilinear Algebra 43 (1997), 87–113. Google Scholar

[13] Tam, T.Y., and Huang, H., An extension of Yamamoto’s theorem on the eigenvalues and singular values of a matrix, Journal of Math. Soc. Japan, 58 (2006), 1197–1202. Google Scholar

[14] Tam, T.Y., Some exponential inequalities for semisimple Lie groups, A chapter of “Operators,Matrices and Analytic Functions”, 539–552, Oper. Theory Adv. Appl. 202, Birkhäuser Verlag, 2010. Google Scholar

[15] Tam, T.Y., A. Horn’s result on matrices with prescribed singular values and eigenvalues, Electron. J. Linear Algebra 21 (2010), 25–27. Google Scholar

[16] von Neumann, J., Some matrix-inequalities and metrization of matric-space, Tomsk. Univ. Rev., 1 (1937), 286–300. Google Scholar

[17] Zhan, X., “Matrix Inequality", Lecture Notes in Mathematics 1790, Springer, Berlin, 2002.Google Scholar

**Received**: 2014-07-04

**Accepted**: 2014-09-14

**Published Online**: 2014-11-07

**Citation Information: **Special Matrices, Volume 2, Issue 1, ISSN (Online) 2300-7451, DOI: https://doi.org/10.2478/spma-2014-0015.

© 2014 Xuhua Liu and Tin-Yau Tam. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.