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

Martin Defant; Marius Junge

Studia Mathematica (1993)

  • Volume: 107, Issue: 1, page 1-14
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Defant, Martin, and Junge, Marius. "Characterization of weak type by the entropy distribution of r-nuclear operators." Studia Mathematica 107.1 (1993): 1-14. <http://eudml.org/doc/216020>.

@article{Defant1993,
abstract = {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.},
author = {Defant, Martin, Junge, Marius},
journal = {Studia Mathematica},
keywords = {entropy numbers; r-nuclear operators; weak type; dual of a Banach space; weak type ; -nuclear operator; Lorentz sequence space},
language = {eng},
number = {1},
pages = {1-14},
title = {Characterization of weak type by the entropy distribution of r-nuclear operators},
url = {http://eudml.org/doc/216020},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Defant, Martin
AU - Junge, Marius
TI - Characterization of weak type by the entropy distribution of r-nuclear operators
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 1
SP - 1
EP - 14
AB - 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.
LA - eng
KW - entropy numbers; r-nuclear operators; weak type; dual of a Banach space; weak type ; -nuclear operator; Lorentz sequence space
UR - http://eudml.org/doc/216020
ER -

References

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  1. [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. 
  2. [CA1] B. Carl, Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), 290-306. Zbl0466.41008
  3. [CA2] B. Carl, Entropy numbers of diagonal operators with applications to eigenvalue problems, J. Approx. Theory 32 (1981), 135-150. Zbl0475.41027
  4. [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. Zbl0509.47017
  5. [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. Zbl0564.47009
  6. [DJ] M. Defant and M. Junge, Some estimates on entropy numbers, Israel J. Math., to appear. Zbl0781.41013
  7. [GEI] S. Geiss, Grothendieck numbers of linear and continuous operators on Banach spaces, Math. Nachr. 110 (1990), 217-230. 
  8. [GKS] Y. Gordon, H. König and C. Schütt, Geometric and probabilistic estimates for entropy and approximation numbers, J. Approx. Theory 49 (1987), 219-239. Zbl0647.47035
  9. [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. 
  10. [KÜH] T. Kühn, Entropy numbers of r-nuclear operators in Banach spaces of type q, Studia Math. 80 (1984), 53-61. Zbl0574.47018
  11. [MA1] V. Mascioni, Weak cotype and weak type in the local theory of Banach spaces, Dissertation, Zürich, 1987. 
  12. [MA2] V. Mascioni, On generalized volume ratio numbers, Bull. Sci. Math. (2) 115 (1991), 483-510. Zbl0771.46010
  13. [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. Zbl0717.46010
  14. [PI1] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., Berlin, 1978, and North-Holland, Amsterdam, 1980. 
  15. [PI2] A. Pietsch, Eigenvalues and s-numbers, Geest & Portig, Leipzig, 1987, and Cambridge University Press, 1987. 
  16. [PS] G. Pisier, The Volume of Convex Bodies and Banach Spaces Geometry, Cambridge University Press, 1989. 
  17. [TOJ] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, Harlow, 1989. Zbl0721.46004

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