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)

 
 

 

$ p$-Convergent sequences and Banach spaces in which $ p$-compact sets are $ q$-compact


Authors: Cándido Piñeiro and Juan Manuel Delgado
Journal: Proc. Amer. Math. Soc. 139 (2011), 957-967
MSC (2010): Primary 46B50, 47B07; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-2010-10508-7
Published electronically: July 28, 2010
MathSciNet review: 2745647
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and investigate the notion of $ p$-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively $ p$-compact if and only if it is contained in the closed convex hull of a $ p$-null sequence. We give a description of the topological dual of the space of all $ p$-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively $ p$-compact subset is relatively $ q$-compact ($ 1\leq q<p$). As an application, Banach spaces satisfying that every relatively $ p$-compact set lies inside the range of a vector measure of bounded variation are characterized.


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

  • 1. J. Bourgain and O. Reinov, On the approximation properties for the space $ H^\infty$, Math. Nachr. 122 (1985), 19-27. MR 871186 (88e:46022)
  • 2. J. M. F. Castillo and F. Sanchez, Dunford-Pettis-like properties of continuous vector function spaces, Rev. Mat. Univ. Complut. Madrid 6 (1993), 43-59. MR 1245024 (95e:46045)
  • 3. J. M. Delgado, C. Piñeiro and E. Serrano, Operators whose adjoints are quasi $ p$-nuclear, Studia Math. 197 (2010), 291-304.
  • 4. -, Density of finite rank operators in the Banach space of $ p$-compact operators, J. Math. Anal. Appl. 370 (2010), 498-505.
  • 5. J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. MR 1342297 (96i:46001)
  • 6. E. Dubinsky, A. Pelczynski and H. P. Rosenthal, On Banach spaces $ X$ for which $ \Pi_2({\mathcal L}_{\infty},X)=B({\mathcal L}_{\infty},X)$, Studia Math. 44 (1972), 617-648. MR 0365097 (51:1350)
  • 7. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoirs Amer. Math. Soc. 16, 1955. MR 0075539 (17:763c)
  • 8. S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193-201. MR 0278090 (43:3822)
  • 9. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0500056 (58:17766)
  • 10. A. Persson and A. Pietsch, $ p$-nukleare und $ p$-integrale Abbildungen in Banachräumen, Studia Math. 33 (1969), 19-62. MR 0243323 (39:4645)
  • 11. A. Pietsch, Operator Ideals, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980. MR 582655 (81j:47001)
  • 12. C. Piñeiro, Sequences in the range of a vector measure of bounded variation, Proc. Amer. Math. Soc. 123 (1995), 3329-3334. MR 1291790 (96a:46033)
  • 13. C. Piñeiro and L. Rodríguez-Piazza, Banach spaces in which every compact lies inside the range of a vector measure, Proc. Amer. Math. Soc. 114 (1992), 507-517. MR 1086342 (92e:46038)
  • 14. O. Reinov, A survey of some results in connection with Grothendieck approximation property, Math. Nachr. 119 (1984), 257-264. MR 774195 (86d:46017)
  • 15. D. P. Sinha and A. K. Karn, Compact operators whose adjoints factor through subspaces of $ \ell_p$, Studia Math. 150 (2002), 17-33. MR 1893422 (2003g:46016)
  • 16. -, Compact operators which factor through subspaces of $ \ell_p$, Math. Nachr. 281 (2008), 412-423. MR 2392123 (2009j:46037)
  • 17. N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, 38, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York, 1989. MR 993774 (90k:46039)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B50, 47B07, 47B10

Retrieve articles in all journals with MSC (2010): 46B50, 47B07, 47B10


Additional Information

Cándido Piñeiro
Affiliation: Department of Mathematics (Faculty of Experimental Sciences), Campus Universitario de El Carmen, Avenida de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Email: candido@uhu.es

Juan Manuel Delgado
Affiliation: Department of Mathematics (Faculty of Experimental Sciences), Campus Universitario de El Carmen, Avenida de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Address at time of publication: Departamento de Matemática Aplicada I (Escuela Técnica Superior de Arquitectura), Avenida de Reina Mercedes, 2, 41012 Sevilla, Spain
Email: jmdelga@us.es

DOI: https://doi.org/10.1090/S0002-9939-2010-10508-7
Keywords: $p$-compact set, $p$-convergent sequence, $p$-nuclear operator, $p$-summing operator, cotype
Received by editor(s): January 21, 2010
Received by editor(s) in revised form: March 6, 2010, and March 22, 2010
Published electronically: July 28, 2010
Additional Notes: This research was supported by MTM2009-14483-C02-01 project (Spain)
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society