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

top
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 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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.