Extensions of Three Matrix Inequalities to Semisimple Lie Groups

Xuhua Liu; Tin-Yau Tam

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 148-154, electronic only
  • ISSN: 2300-7451

Abstract

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

How to cite

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Xuhua Liu, and Tin-Yau Tam. "Extensions of Three Matrix Inequalities to Semisimple Lie Groups." Special Matrices 2.1 (2014): 148-154, electronic only. <http://eudml.org/doc/267514>.

@article{XuhuaLiu2014,
abstract = {We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.},
author = {Xuhua Liu, Tin-Yau Tam},
journal = {Special Matrices},
keywords = {Araki-Lieb-Thirring inequality; positive definite matrices; semisimple Lie groups; log majorization; Kostant’s pre-order; Kostant's pre-order},
language = {eng},
number = {1},
pages = {148-154, electronic only},
title = {Extensions of Three Matrix Inequalities to Semisimple Lie Groups},
url = {http://eudml.org/doc/267514},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Xuhua Liu
AU - Tin-Yau Tam
TI - Extensions of Three Matrix Inequalities to Semisimple Lie Groups
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 148
EP - 154, electronic only
AB - We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.
LA - eng
KW - Araki-Lieb-Thirring inequality; positive definite matrices; semisimple Lie groups; log majorization; Kostant’s pre-order; Kostant's pre-order
UR - http://eudml.org/doc/267514
ER -

References

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  1. [1] Araki, H., On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), 167–170. Zbl0705.47020
  2. [2] Audenaert, K. M. R., On the Araki-Lieb-Thirring Inequality, Int. J. Inf. Syst. Sci. 4 (2008), 78–83. Zbl1156.15008
  3. [3] Bhatia, R., “Matrix Analysis", Springer-Verlag, New Yor, 1997. Zbl0863.15001
  4. [4] Helgason, S., “Differential Geometry, Lie Groups, and Symmetric Spaces”, Academic Press, 1978. Zbl0451.53038
  5. [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. Zbl0885.47003
  6. [6] Horn, A., Doubly stochastic matrices and the diagonal of a rotation of matrix, Amer. J. Math. 76 (1954), 620–630. Zbl0055.24601
  7. [7] Knapp, A. W., “Lie Groups beyond an Introduction", 2nd ed., Birkhäuser, 2002. Zbl1075.22501
  8. [8] Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455. Zbl0293.22019
  9. [9] Lieb, E., and Thirring,W., in Studies inMathematical Physics (Eds. E. Lieb, B. Simon and A.Wightman), p.301–302, Princeton Press, 1976. 
  10. [10] Marshall, A. W., I. Olkin, and B. C. Arnold, “Inequalities: Theory of Majorization and its Applications (2nd ed.)”, Springer, 2011. Zbl1219.26003
  11. [11] Simon, B., “Trace Ideals and Their Applications”, London Mathematical Society Lecture Note Series, 35, Cambridge Univ. Press, 1979. Zbl0423.47001
  12. [12] Tam, T.Y., Kostant’s convexity theorem and the compact classical groups, Linear and Multilinear Algebra 43 (1997), 87–113. Zbl0894.22003
  13. [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. Zbl1117.15008
  14. [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. Zbl1192.15008
  15. [15] Tam, T.Y., A. Horn’s result on matrices with prescribed singular values and eigenvalues, Electron. J. Linear Algebra 21 (2010), 25–27. Zbl1215.15022
  16. [16] von Neumann, J., Some matrix-inequalities and metrization of matric-space, Tomsk. Univ. Rev., 1 (1937), 286–300. 
  17. [17] Zhan, X., “Matrix Inequality", Lecture Notes in Mathematics 1790, Springer, Berlin, 2002. 

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