Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Almost compactness and decomposability of integral operators

Authors: Walter Schachermayer and Lutz Weis
Journal: Proc. Amer. Math. Soc. 81 (1981), 595-599
MSC: Primary 47G05; Secondary 45P05, 47B05, 47B38
MathSciNet review: 601737
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (X,\mu )$, $ (Y,v)$ be finite measure spaces and $ 1 < q \leqslant \infty $, $ 1 \leqslant p \leqslant q$. An integral operator $ \operatorname{Int}(k):{L^q}(v) \to {L^p}(\mu )$ becomes compact, if we cut away a suitably chosen subset of $ X$ of arbitrarily small measure. As a consequence we prove that $ \operatorname{Int}(k)$ may be written as the sum of a Carleman operator and an orderbounded integral operator, where the orderbounded part may be chosen to be compact and of arbitrarily small norm.

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

  • [1] T. Ando, On compactness of integral operators, Indag. Math. 24 (1962), 235-239. MR 0139016 (25:2456)
  • [2] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R.I., 1977. MR 0453964 (56:12216)
  • [3] P. G. Dodds, Compact kernel operators on Banach function spaces, (preprint).
  • [4] N. Dunford, Integration and linear operations, Trans. Amer. Math. Soc. 40 (1936), 474-494. MR 1501886
  • [5] N. E. Gretsky and J. J. Uhl, Carleman and Korotkov operators on Banach spaces, (preprint). MR 621370 (82i:47044)
  • [6] A. Grothendieck, Produits tensoriels topologiques, Mem. Amer. Math. Soc. No. 16 (1955). MR 0075539 (17:763c)
  • [7] P. R. Halmos and V. Sunder, Bounded integral operators on $ {L^2}$ spaces, Ergebnisse der Math., Bd. 96, Springer-Verlag, Berlin and New York, 1978. MR 517709 (80g:47036)
  • [8] V. Korotkov, On some properties of partially integral operators, Dokl. Akad. Nauk SSSR 217 (1974); English transl. in Soviet Math. Dokl. 15 (1974), 1114-1117. MR 0361934 (50:14376)
  • [9] M. A. Krasnosel'skiĭ et al., Integral operators in spaces of summable functions, Nordhoff, Groningen, 1976.
  • [10] B. Maurey, Théorèmes de factorisation pour les operateurs linéaires à valeurs dans $ {L^p}$, Asterisque 11, Paris, 1974.
  • [11] E. M. Nikišin, Resonance theorems and superlinear operators, Upehi Mat. Nauk 25 (1970), 125-191; English transl. in Russian Math. Surveys 25 (1970), 125-197. MR 0296584 (45:5643)
  • [12] W. Schachermayer, Integral operators on $ {L^p}$ spaces, (preprint).
  • [13] C. Stegall, The Radon-Nikodym property in conjugate Banach spaces. II, Trans. Amer. Math. Soc. (to appear) MR 603779 (82k:46030)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47G05, 45P05, 47B05, 47B38

Retrieve articles in all journals with MSC: 47G05, 45P05, 47B05, 47B38

Additional Information

Keywords: Integral operator
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society