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)

Request Permissions   Purchase Content 
 
 

 

The ideal of $ p$-compact operators and its maximal hull


Author: Albrecht Pietsch
Journal: Proc. Amer. Math. Soc. 142 (2014), 519-530
MSC (2010): Primary 47B07, 47B10; Secondary 46B22, 47L20
DOI: https://doi.org/10.1090/S0002-9939-2013-11758-2
Published electronically: October 17, 2013
MathSciNet review: 3133993
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many recent papers have been devoted to the ideal of $ p$-compact operators, denoted by $ \mathfrak{K}_p$. In this note, we will revisit their main results by very simple proofs that are based on the general theory of operator ideals as developed in the author's monograph more than 30 years ago. In several cases the outcome is even better. Moreover, the ideal of all operators with $ p$-summing duals is identified as the maximal hull of $ \mathfrak{K}_p$. As a byproduct, we show that $ \mathfrak{K}_p^{\rm max}$ is stable under the formation of projective tensor products.


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

  • [1] Kati Ain, Rauni Lillemets, and Eve Oja, Compact operators which are defined by $ \ell _p$-spaces, Quaest. Math. 35 (2012), no. 2, 145-159. MR 2945718, https://doi.org/10.2989/16073606.2012.696819
  • [2] B. Carl, A. Defant, and M. S. Ramanujan, On tensor stable operator ideals, Michigan Math. J. 36 (1989), no. 1, 63-75. MR 989937 (90b:47080), https://doi.org/10.1307/mmj/1029003882
  • [3] Joel S. Cohen, Absolutely $ p$-summing, $ p$-nuclear operators and their conjugates, Math. Ann. 201 (1973), 177-200. MR 0370245 (51 #6472)
  • [4] J. M. Delgado, C. Piñeiro, and E. Serrano, Operators whose adjoints are quasi $ p$-nuclear, Studia Math. 197 (2010), no. 3, 291-304. MR 2607494 (2011g:47041), https://doi.org/10.4064/sm197-3-6
  • [5] Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438 (94e:46130)
  • [6] Joe Diestel, Jan H. Fourie, and Johan Swart, The metric theory of tensor products, Grothendieck's résumé revisited, American Mathematical Society, Providence, RI, 2008. MR 2428264 (2010a:46005)
  • [7] Jan Fourie and Johan Swart, Banach ideals of $ p$-compact operators, Manuscripta Math. 26 (1978/79), no. 4, 349-362. MR 520105 (80b:47059), https://doi.org/10.1007/BF01170259
  • [8] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp. (French). MR 0075539 (17,763c)
  • [9] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1-79 (French). MR 0094682 (20 #1194)
  • [10] J. R. Holub, Tensor product mappings, Math. Ann. 188 (1970), 1-12. MR 0284828 (44 #2052)
  • [11] J. R. Holub, Tensor product mappings. II, Proc. Amer. Math. Soc. 42 (1974), 437-441. MR 0331104 (48 #9438)
  • [12] Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45-90 (French). MR 0443015 (56 #1388)
  • [13] Eve Oja, A remark on the approximation of $ p$-compact operators by finite-rank operators , J. Math. Anal. Appl. 387 (2012), no. 2, 949-952. MR 2853187 (2012i:46025), https://doi.org/10.1016/j.jmaa.2011.09.055
  • [14] Arne Persson and Albrecht Pietsch, $ p$-nukleare und $ p$-integrale Abbildungen in Banachräumen, Studia Math. 33 (1969), 19-62 (German). MR 0243323 (39 #4645)
  • [15] Albrecht Pietsch, Operator ideals, Mathematische Monographien [Mathematical Monographs], vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 519680 (81a:47002)
  • [16] Cándido Piñeiro and Juan Manuel Delgado, $ p$-convergent sequences and Banach spaces in which $ p$-compact sets are $ q$-compact, Proc. Amer. Math. Soc. 139 (2011), no. 3, 957-967. MR 2745647 (2011j:46031), https://doi.org/10.1090/S0002-9939-2010-10508-7
  • [17] -, $ p$-Compact sets and $ p$-summing evaluation maps, preprint.
  • [18] O. I. Reinov, On linear operators with $ p$-nuclear adjoints, Vestnik St. Petersburg Univ. Math. 33 (2000), no. 4, 19-21 (2001). MR 1868256 (2002i:47033)
  • [19] Deba P. Sinha and Anil K. Karn, Compact operators whose adjoints factor through subspaces of $ l_p$, Studia Math. 150 (2002), no. 1, 17-33. MR 1893422 (2003g:46016), https://doi.org/10.4064/sm150-1-3
  • [20] Deba Prasad Sinha and Anil Kumar Karn, Compact operators which factor through subspaces of $ l_p$, Math. Nachr. 281 (2008), no. 3, 412-423. MR 2392123 (2009j:46037), https://doi.org/10.1002/mana.200510612
  • [21] Irmtraud Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banach-Räume und ihre Erzeugung, Beiträge Anal. 5 (1973), 75-89 (German). MR 0353030 (50 #5516)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B07, 47B10, 46B22, 47L20

Retrieve articles in all journals with MSC (2010): 47B07, 47B10, 46B22, 47L20


Additional Information

Albrecht Pietsch
Affiliation: Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Address at time of publication: Biberweg 7, D-07749 Jena, Germany
Email: a.pietsch@uni-jena.de

DOI: https://doi.org/10.1090/S0002-9939-2013-11758-2
Keywords: $p$-compact operators, quasi $p$-nuclear operators, absolutely $p$-summing operators, $(s,p)$-mixing operators, maximal operator ideals, surjective operator ideals
Received by editor(s): July 11, 2011
Received by editor(s) in revised form: March 4, 2012, and March 18, 2012
Published electronically: October 17, 2013
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society