Abstract

We obtain a representation of unitarily invariant norm in terms of Ky Fan norms [see R. Bhatia, Matrix Analysis, Springer, New York et al., 1997, p. 35]. Indeed we obtain a more general result in the context of Eaton triple with reduced triple. Examples are given.

Author information

Contact details are reproduced from the original publication and may be historical.

Piotr Niemiec

Jagiellonian University, Institute of Mathematics, ul. Łojasiewicza 6, 30-348 Kraków, Poland

piotr.niemiec@uj.edu.pl

Tin-Yau Tam

Dept. of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, U.S.A.

tamtiny@auburn.edu

Keywords

  • Eaton triple
  • G-invariant norms
  • symmetric gauge functions
  • Ky Fan norms
  • finite reflection groups

Mathematics Subject Classification

15A60, 65F35

References

  1. R. Bhatia: Matrix Analysis, Springer, New York (1997).https://doi.org/10.1007/978-1-4612-0653-8Google Scholar
  2. N. Bourbaki: Elements of Mathematics. Lie Groups and Lie Algebras. Chapter 4–6, Springer, Berlin (2002).https://doi.org/10.1007/978-3-540-89394-3Google Scholar
  3. M. L. Eaton, M. D. Perlman: Reflection groups, generalized Schur functions and the geometry of majorization, Ann. Probab. 5 (1977) 829–860.https://doi.org/10.1214/aop/1176995655Google Scholar
  4. L. C. Grove, C. T. Benson: Finite Reflection Groups, 2nd Ed., Springer, New York (1985).https://doi.org/10.1007/978-1-4757-1869-0Google Scholar
  5. R. R. Holmes, T. Y. Tam: Distance to the convex hull of an orbit under the action of a compact group, J. Aust. Math. Soc., Ser. A 66 (1999) 331–357.https://doi.org/10.1017/S1446788700036648Google Scholar
  6. R. Horn, R. Mathias: Cauchy-Schwarz inequalities associated with positive semidefinite matrices, Linear Algebra Appl. 142 (1990) 63–82.https://doi.org/10.1016/0024-3795(90)90256-CGoogle Scholar
  7. J. E. Humphreys: Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511623646Google Scholar
  8. R. Kane: Reflection Groups and Invariant Theory, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3542-0Google Scholar
  9. A. S. Lewis: Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl. 17 (1996) 927–949.https://doi.org/10.1137/S0895479895283173Google Scholar
  10. C. K. Li, N. K. Tsing: G-invariant norms and G(c)-radii, Linear Algebra Appl. 150 (1991) 179–194.https://doi.org/10.1016/0024-3795(91)90168-VGoogle Scholar
  11. J. von Neumann: Some matrix-inequalities and metrization of matric-space, Mitt. Forsch.-Inst. Math. Mech. Univ. Tomsk 1 (1937) 286–300; in: Collected Works. Vol. IV. Continuous Geometry and Other Topics, A. H. Taub (ed.), Pergamon Press, Oxford (1962) 205–219.Google Scholar
  12. M. Niezgoda: Group majorization and Schur type inequalities, Linear Algebra Appl. 268 (1998) 9–30.https://doi.org/10.1016/S0024-3795(97)89322-6Google Scholar
  13. M. Niezgoda: An analytical characterization of effective and of irreducible groups inducing cone orderings, Linear Algebra Appl. 269 (1998) 105–114.https://doi.org/10.1016/S0024-3795(97)00060-8Google Scholar
  14. M. Niezgoda, T. Y. Tam: On norm property of G(c)-radii and Eaton triples, Linear Algebra Appl. 336 (2001) 119–130.https://doi.org/10.1016/S0024-3795(01)00310-XGoogle Scholar
  15. W. Rudin: Functional Analysis, McGraw-Hill, New York (1991).Google Scholar
  16. T. Y. Tam: An extension of a result of Lewis, Electron. J. Linear Algebra 5 (1999) 1–10.https://doi.org/10.13001/1081-3810.1026Google Scholar
  17. T. Y. Tam: Group majorization, Eaton triples and numerical range, Linear Multilinear Algebra 47 (2000) 11–28.https://doi.org/10.1080/03081080008818628Google Scholar
  18. T. Y. Tam: Generalized Schur-concave functions and Eaton triples, Linear Multilinear Algebra 50 (2002) 113–120.https://doi.org/10.1080/03081080290019612Google Scholar
  19. T. Y. Tam, W. C. Hill: On G-invariant norms, Linear Algebra Appl. 331 (2001) 101–112.https://doi.org/10.1016/S0024-3795(01)00270-1Google Scholar

Suggested citation

P. Niemiec, T.-Y. Tam. “A Representation of G-invariant norms for Eaton Triple.” Journal of Convex Analysis 18 (2011), No. 1, 59–65. https://doi.org/10.68381/jca18003

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.