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A structure theorem for the polars of unitarily invariant norms

Authors: Govind S. Mudholkar and Marshall Freimer
Journal: Proc. Amer. Math. Soc. 95 (1985), 331-337
MSC: Primary 15A60; Secondary 47A30
MathSciNet review: 806065
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Abstract: The unitarily invariant norms of matrices, or operators, are essentially the symmetric norms of their singular values. A subclass of these norms depending upon only a few largest of the singular values is considered, and the polars of these norms are characterized. The result is then used to obtain generalizations of some well-known inequalities. The implications for operators on infinite-dimensional spaces are discussed.

References [Enhancements On Off] (What's this?)

  • [1] R. Bhatia, Analysis of spectral variation and some inequalities, Trans. Amer. Math. Soc. 272 (1982), 323-331. MR 656492 (83k:15015)
  • [2] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760-766. MR 0045952 (13:661e)
  • [3] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear non-self adjoint operators, Amer. Math. Soc. Transl. 18 (1969). MR 0246142 (39:7447)
  • [4] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, 1934.
  • [5] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. II, Springer-Verlag, Berlin, 1970. MR 551496 (81k:43001)
  • [6] R. Kunze, $ {L^p}$-Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519-540. MR 0100235 (20:6668)
  • [7] C. A. McCarthy, $ {c_p}$, Israel J. Math. 5 (1967), 249-271. MR 0225140 (37:735)
  • [8] G. S. Mudholkar, M. Freimer and P. Subbaiah, An extension of Hölder's inequality, J. Math. Anal. Appl. 102 (1984), 435-441. MR 755974 (85m:26023)
  • [9] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970. MR 0274683 (43:445)
  • [10] R. Schatten, A theory of cross-spaces, Ann. of Math. Studies, No. 26, Princeton Univ. Press, Princeton, N.J., 1950. MR 0036935 (12:186e)
  • [11] -, Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1960. MR 0119112 (22:9878)
  • [12] B. Simon, Trace ideals and their applications, Cambridge Univ. Press, 1979. MR 541149 (80k:47048)
  • [13] J. von Neumann, Some matrix inequalities and metrization of matric spaces, Tomsk. Univ. Rev. 1 (1937), 286-300 (also Collected Works, Vol. IV).

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Keywords: Symmetric gauge functions, Holder's inequality, conjugates, $ {c_p}$
Article copyright: © Copyright 1985 American Mathematical Society

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