ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
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Complex interpolation of compact operators mapping into lattice couples; pp. 19–28
PDF | doi: 10.3176/proc.2010.1.04

Author
Michael Cwikel
Abstract
After 44 years it is still not known whether an operator mapping one Banach couple boundedly into another and acting compactly on one (or even both) of the “endpoint” spaces also acts compactly between the complex interpolation spaces generated by these couples. We answer this question affirmatively in certain cases where the “range” Banach couple is a couple of lattices on the same measure space.
References

  1. Bartle, R. G. On compactness in functional analysis. Trans. Amer. Math. Soc., 1955, 79, 35–57.
doi:10.2307/1992835

  2. Bergh, J. On the relation between the two complex methods of interpolation. Indiana Univ. Math. J., 1979, 28, 775–778.
doi:10.1512/iumj.1979.28.28054

  3. Calderón, A. P. Intermediate spaces and interpolation, the complex method. Studia Math., 1964, 24, 113–190.

  4. Cobos, F., Kühn, T., and Schonbek, T. One-sided compactness results for Aronszajn-Gagliardo functors. J. Funct. Anal., 1992, 106, 274–313.
doi:10.1016/0022-1236(92)90049-O

  5. Cwikel, M. Real and complex interpolation and extrapolation of compact operators. Duke Math. J., 1992, 65, 333–343.
doi:10.1215/S0012-7094-92-06514-8

  6. Cwikel, M. Lecture notes on duality and interpolation spaces. arXiv:0803.3558 [math.FA].

  7. Cwikel, M. and Janson, S. Complex interpolation of compact operators mapping into the couple (FLFL1). In Contemporary Mathematics, Vol. 445 (De Carli, L. and Milman, M., eds). American Mathematical Society, Providence R. I., 2007, 71–92.

  8. Cwikel, M. and Kalton, N. J. Interpolation of compact operators by the methods of Calderón and Gustavsson–Peetre. Proc. Edinburgh Math. Soc., 1995, 38, 261–276.
doi:10.1017/S0013091500019076

  9. Cwikel, M., Krugljak, N., and Mastyło, M. On complex interpolation of compact operators. Illinois J. Math., 1996, 40, 353–364.

10. Cwikel, M. and Levy, E. Estimates for covering numbers in Schauder’s theorem about adjoints of compact operators. arXiv:0810.4240 [math.FA].

11. Cwikel, M. and Nilsson, P. The coincidence of real and complex interpolation methods for couples of weighted Banach lattices. In Proceedings of a Conference on Interpolation Spaces and Allied Topics in Analysis, Lund, 1983 (Cwikel, M. and Peetre, J., eds). Lecture Notes in Mathematics, 1070. Springer, Berlin–Heidelberg–New York–Tokyo, 1984, 54–65.

12. Cwikel, M. and Nilsson, P. G. Interpolation of weighted Banach lattices. Mem. Amer. Math. Soc., 2003, 165(787).

13. Dunford, N. and Schwartz, J. T. Linear Operators. Part 1: General Theory. Interscience Publishers, New York, 1958.

14. Kakutani, S. A proof of Schauder’s theorem. J. Math. Soc. Japan, 1951, 3, 228–231.
doi:10.2969/jmsj/00310228

15. Krein, S. G., Petunin, Ju. I., and Semenov, E. M. Interpolation of Linear Operators. Translations of Mathematical Monographs, Vol. 54. American Mathematical Society, Providence R.I., 1982.

16. Levy, E. Weakly Compact “Matrices”, Fubini-Like Property and Extension of Densely Defined Semigroups of Operators. arXiv:0704.3558v3 [math.FA].

17. Lindenstrauss, J. and Tzafriri, L. Classical Banach Spaces II. Ergebnisse der Mathematik und ihrer Grenzgebiete 97. Springer, Berlin–Heidelberg–New York, 1979.

18. Lozanovskii, G. Ya. On some Banach lattices. Sibirsk. Matem. Zh., 1969, 10, 584–597 (in Russian); Siberian Math. J., 1969, 10, 419–431.
doi:10.1007/BF01078332

19. Mujica, J. The Kakutani’s precompactness lemma. J. Math. Anal. Appl., 2004, 297, 477–489.
doi:10.1016/j.jmaa.2004.03.070

20. Phillips, R. S. On weakly compact subsets of a Banach space. Amer. J. Math., 1943, 65, 108–136.
doi:10.2307/2371776

21. Pustylnik, E. Interpolation of compact operators in spaces of measurable functions. Math. Ineq. Appl., 2008, 11, 467–476.

22. Reisner, S. On two theorems of Lozanovskii concerning intermediate Banach lattices. In Geometrical Aspects of Functional Analysis – Israel Seminar 1986/87. Lecture Notes in Mathematics, 1317. Springer, Berlin–Heidelberg–New York–Tokyo, 1988, 67–83.

23. Šmulian, V. Sur les ensembles compacts et faiblement compacts dans l’espace du type (B). Rec. Math. (Mat. Sbornik), 1943, 12(54), 91–97.

24. Zaanen, A. C. Integration. North Holland, Amsterdam, 1967.
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