Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T09:39:19.357Z Has data issue: false hasContentIssue false
  • English
  • Français

Interpolation of compact operators by the methods of Calderón and Gustavsson–Peetre

Published online by Cambridge University Press:  20 January 2009

M. Cwikel  and
N. J. Kalton
M. Cwikel
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 3200, Israel
N. J. Kalton
Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X = (X0, X1) and Y = (Y0, Y1) be Banach couples and suppose T:X→Y is a linear operator such that T:X0→Y0 is compact. We consider the question whether the operator T:[X0, X1]θ→[Y0, Y1]θ is compact and show a positive answer under a variety of conditions. For example it suffices that X0 be a UMD-space or that X0 is reflexive and there is a Banach space so that X0 = [W, X1]α, for some 0<α<1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 2 , June 1995 , pp. 261 - 276
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Bennett, C. and Sharpley, R., Interpolation of Operators (Academic Press, New York 1988).Google Scholar
2.Bergh, J., On the relation between the two complex methods of interpolation, Indiana Univ. Math.J. 28 (1979), 775–778.CrossRefGoogle Scholar
3.Bergh, J. and Löfström, J., Interpolation spaces (An Introduction, Springer, Berlin 1976).Google Scholar
4.Bourgain, J., Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163–168.CrossRefGoogle Scholar
5.Brudnyi, Y. and Krugljak, N., Interpolation functors and interpolation spaces, Volume 1 (North-Holland, Amsterdam 1991).Google Scholar
6.Burkholder, D. L., A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.CrossRefGoogle Scholar
7.Burkholder, D. L., A geometric condition that implies the existence of certain singular integrals of Banach space-valued functions, in Conference on harmonic analysis in honor of A. Zygmund (Beckner, W., Calderón, A. P., Fefferman, R., Jones, P. W., editors, Wadsworth, Belmont, Califronia, 1983), 270–286.Google Scholar
8.Calderón, A. P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.CrossRefGoogle Scholar
9.Cobos, F., Kühn, T. and Schonbek, T., One-sided compactness results for Aronszajn-Gagliardo functors, J, Funct. Anal. 106 (1992), 274–313.Google Scholar
10.Cobos, F. and Peetre, J., Interpolation of compactness using Aronszajn-Gagliardo functors, Israel J. Math. 68 (1989), 220–240.Google Scholar
11.Cwikel, M., Complex interpolation, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), 1005–1009.Google Scholar
12.Cwikel, M., Real and complex interpolation and extrapolation of compact operators. Duke. Math. J. 65 (1992), 333–343.CrossRefGoogle Scholar
13.Gustavsson, J. and Peetre, J., Interpolation of Orlicz spaces, Studia Math. 60 (1977), 33–59.CrossRefGoogle Scholar
14.Gutierrez, J. A., On the boundedness of the Banach space-valued Hilbert transform (Ph.D. thesis, University of Texas, Austin 1982).Google Scholar
15.Hayakawa, K., Interpolation by the real method preserves compactness of operators, J. Math. Soc. Japan 21 (1969), 189–199.Google Scholar
16.Janson, S., Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981), 50–73.CrossRefGoogle Scholar
17.Krasnosel'skii, M. A., On a theorem of M. Reisz, Soviet Math. Dokl. 1 (1960), 229–231.Google Scholar
18.Lions, J. L. and Peetre, J., Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19 (1964), 5–68.CrossRefGoogle Scholar
19.Mastylo, M., On interpolation of compact operators, preprint.Google Scholar
20.Peetre, J., Sur l'utilization des suites inconditionallement sommables dan la theorie des espaces d'interpolation, Rend. Sem. Mat. Univ. Padova 46 (1971), 173–190.Google Scholar
21.Persson, A., Compact linear mappings between interpolation spaces. Ark Mat. 5 (1964), 215–219.Google Scholar
22.Ovchinnokov, V. I., The method of orbits in interpolation theory, in Mathematical Reports, Vol. 1, Part 2 (Harwood Academic Publishers 1984), 349–516.Google Scholar