Advertisement

Convex Spectral Functions of Compact Operators, Part II: Lower Semicontinuity and Rearrangement Invariance

  • Jonathan M. Borwein
  • Adrian S. Lewis
  • Qiji J. Zhu
Part of the Applied Optimization book series (APOP, volume 47)

Abstract

It was shown in Part I of this work that the Gateaux differentiability of a convex unitarily invariant function is characterized by that of a similar induced rearrangement invariant function on the corresponding spectral space. A natural question is then whether this is also the case for Fréchet differentibility. In this paper we show the answer is positive. Although the result appears very natural, the proof turns out to be quite technically involved.

Key words

Convex spectral functions differentiability rearrangement invariant functions 

Mathematics Subject Classification (2000)

49J52 47B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arazy, J. (1981), On the geometry of the unit ball of unitary matrix spaces, Integral Equations and Operator theory, 4, 151–171.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bhatia, R. (1997), Matrix Analysis, Springer, New York.CrossRefGoogle Scholar
  3. Borwein, J. M., Read, J., Lewis, A. S. and Zhu, Q. J. (1999), Convex spectral functions of compact operators, International J. of Nonlinear and Convex Analysis, 1, 17–35.MathSciNetGoogle Scholar
  4. Borwein, J. M. and Zhu, Q. J. (1999), A survey of subdifferentials and their applications, CECM Research Report 98–105 (1998), Nonlinear Analysis, TMA, 38, 687–773.MathSciNetzbMATHGoogle Scholar
  5. Clarke, F. H. (1990), Optimization and Nonsmooth Analysis, John Wiley amp; Sons, New York, 1983, Russian edition MIR, Moscow, (1988). Reprinted as Vol. 5 of the series Classics in Applied Mathematics, SIAM, Philadelphia.Google Scholar
  6. Faybusovich, L. (1998), Infinite-dimensional semidefinite programming: regularized determinants and self-concordant barriers, Topics in semidefinite and interior-point methods (Toronto, ON, 1996), 39–49, Fields Inst. Commun., 18, Amer. Math. Soc. Providence, RI.Google Scholar
  7. Friedland, S. and Nowosad, P. (1981), Extremal eigenvalue problems with indefinite kernels, Adv. in Math., 40,128–154.Google Scholar
  8. Gohberg, I. C. and Krein, M. G. (1969), Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc. Providence, RI.Google Scholar
  9. Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952), Inequalities, Cambridge University Press, Cambridge, U. K.Google Scholar
  10. Lennard, C. (1990), C1 is uniformly Kadec-Klee. Proc. Amer. Math. Soc. 109, No. 1, 71–77.MathSciNetzbMATHGoogle Scholar
  11. Lewis, A. S. (1996), Convex analysis on the Hermitian matrices, SIAM J. Op-tim., 6, 164–177.zbMATHCrossRefGoogle Scholar
  12. Lewis, A. S. (1999), Nonsmooth analysis of eigenvalues, Mathematical Programming, 84, 1–24.MathSciNetzbMATHGoogle Scholar
  13. Lewis, A. S. (1999), Lidskii’s theorem via nonsmooth analysis, SIAM J. Matrix An., Vol. 21, 379–381.zbMATHCrossRefGoogle Scholar
  14. Markus, A. S. (1964), The eigen-and singular values of the sum and product of linear operators, Uspehi Mat. Nauk, 9, 91–120.MathSciNetGoogle Scholar
  15. Minc, H. (1988), Nonnegative Matrices, John Wiley amp; Sons, New York.Google Scholar
  16. Pederson, G. (1989), Analysis Now, Springer Verlag, Berlin.CrossRefGoogle Scholar
  17. Phelps, R. R. (1993), Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, No. 1364, Springer Verlag, N.Y., Berlin, Tokyo, Second Edition.Google Scholar
  18. Read, J. (1996), The approximation of optimal control of vibrations–a geometrical method, Mathematical Methods in the Applied Sciences, 19, 87–129.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Rockafellar, R. T. (1970), Convex Analysis, Princeton University Press, Princeton, N.J.zbMATHGoogle Scholar
  20. Simon, B. (1979), Trace Ideals and Their Applications,Cambridge University Press.Google Scholar
  21. von Neumann, J. (1937), Some matrix inequalities and metrization of matricspace, Tomsk University Review, 1 (1937) 286–300. In: Collected Works, Pergamon, Oxford, 1962, Vol. IV, 205–218.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Jonathan M. Borwein
    • 1
  • Adrian S. Lewis
    • 2
  • Qiji J. Zhu
    • 3
  1. 1.Centre for Experimental and Constructive Mathematics, Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Department of Mathematics and StatisticsWestern Michigan UniversityKalamazooUSA

Personalised recommendations