headerpos: 12198
 
 
 

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
Publisher
Journal Information
» Editorial Board
» Editorial Policy
» Archival Policy
» Article Publication Charges
» Copyright and Licensing Policy
Guidelines for Authors
» For Authors
» Instructions to Authors
» LaTex style files
Guidelines for Reviewers
» For Reviewers
» Review Form
Open Access
List of Issues
» 2019
» 2018
» 2017
» 2016
» 2015
» 2014
» 2013
» 2012
» 2011
» 2010
Vol. 59, Issue 4
Vol. 59, Issue 3
Vol. 59, Issue 2
Vol. 59, Issue 1
» 2009
» 2008
» Back Issues Phys. Math.
» Back Issues Chemistry
» Back issues (full texts)
  in Google. Phys. Math.
» Back issues (full texts)
  in Google. Chemistry
» Back issues (full texts)
  in Google Engineering
» Back issues (full texts)
  in Google Ecology
» Back issues in ETERA Füüsika, Matemaatika jt
Subscription Information
» Prices
Internet Links
Support & Contact
Publisher
» Staff
» Other journals

Complex interpolation of compact operators mapping into lattice couples; pp. 19–28

(Full article in PDF format) doi: 10.3176/proc.2010.1.04


Authors

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.

Keywords

functional analysis, complex interpolation, compact operator, Banach lattice.

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.
 
Back

Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December