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1993 | 107 | 1 | 1-14
Tytuł artykułu

Characterization of weak type by the entropy distribution of r-nuclear operators

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EN
Abstrakty
EN
The dual of a Banach space X is of weak type p if and only if the entropy numbers of an r-nuclear operator with values in a Banach space of weak type q belong to the Lorentz sequence space $ℓ_{s,r}$ with 1/s + 1/p + 1/q = 1 + 1/r (0 < r < 1, 1 ≤ p, q ≤ 2). It is enough to test this for Y = X*. This extends results of Carl, König and Kühn.
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autor
  • Mathematisches Seminar, Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
Bibliografia
  • [BPST] J. Bourgain, A. Pajor, S. J. Szarek and N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, in: Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 1987-88, Lecture Notes in Math. 1376, Springer, 1989, 50-63.
  • [CA1] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), 290-306.
  • [CA2] B. Carl, Entropy numbers of diagonal operators with applications to eigenvalue problems, J. Approx. Theory 32 (1981), 135-150.
  • [CA3] B. Carl, On a characterization of operators from $l_q$ into a Banach space of type p with some applications to eigenvalue problems, J. Funct. Anal. 48 (1982), 394-407.
  • [CA4] B. Carl, Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces, Ann. Inst. Fourier (Grenoble) 35 (3) (1985), 79-118.
  • [DJ] M. Defant and M. Junge, Some estimates on entropy numbers, Israel J. Math., to appear.
  • [GEI] S. Geiss, Grothendieck numbers of linear and continuous operators on Banach spaces, Math. Nachr. 110 (1990), 217-230.
  • [GKS] Y. Gordon, H. König and C. Schütt, Geometric and probabilistic estimates for entropy and approximation numbers, J. Approx. Theory 49 (1987), 219-239.
  • [KÖN] H. König, Eigenvalues of p-nuclear operators, in: Proc. Internat. Conf. Operator Algebras, Ideals, and Their Applications in Theoretical Physics, H. Baumgärtel et al. (eds.), Teubner, Leipzig, 1978, 106-113.
  • [KÜH] T. Kühn, Entropy numbers of r-nuclear operators in Banach spaces of type q, Studia Math. 80 (1984), 53-61.
  • [MA1] V. Mascioni, Weak cotype and weak type in the local theory of Banach spaces, Dissertation, Zürich, 1987.
  • [MA2] V. Mascioni, On generalized volume ratio numbers, Bull. Sci. Math. (2) 115 (1991), 483-510.
  • [PTJ] A. Pajor and N. Tomczak-Jaegermann, Volume ratio and other s-numbers of operators related to local properties of Banach spaces, J. Funct. Anal. 87 (1989), 273-279.
  • [PI1] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., Berlin, 1978, and North-Holland, Amsterdam, 1980.
  • [PI2] A. Pietsch, Eigenvalues and s-numbers, Geest & Portig, Leipzig, 1987, and Cambridge University Press, 1987.
  • [PS] G. Pisier, The Volume of Convex Bodies and Banach Spaces Geometry, Cambridge University Press, 1989.
  • [TOJ] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, Harlow, 1989.
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Bibliografia
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