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Abstract:

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on KA, the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on KA. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with KA instead of A. We exemplify the power of the Carl-Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness. © 2013 Elsevier Inc.

Registro:

Documento: Artículo
Título:The Banach ideal of A-compact operators and related approximation properties
Autor:Lassalle, S.; Turco, P.
Filiación:Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, B1644BID, Victoria, Buenos Aires, Argentina
IMAS - CONICET, Argentina
Departamento de Matemática - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina
Palabras clave:Approximation properties; Compact sets; Operator ideals
Año:2013
Volumen:265
Número:10
Página de inicio:2452
Página de fin:2464
DOI: http://dx.doi.org/10.1016/j.jfa.2013.07.001
Título revista:Journal of Functional Analysis
Título revista abreviado:J. Funct. Anal.
ISSN:00221236
CODEN:JFUAA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221236_v265_n10_p2452_Lassalle

Referencias:

  • Aron, R., Maestre, M., Rueda, P., P-Compact holomorphic mappings (2010) RACSAM, 104 (2), pp. 353-364
  • Aron, R., Schottenloher, M., Compact holomorphic mappings and the approximation property (1976) J. Funct. Anal., 21, pp. 7-30
  • Carl, B., Stephani, I., On A-compact operators, generalized entropy numbers and entropy ideals (1984) Math. Nachr., 199, pp. 77-95
  • Casazza, P.G., Approximation properties (2001) Handbook of the Geometry of Banach Spaces, vol. I, pp. 271-316. , North-Holland Publishing Co., Amsterdam
  • Choi, Y.S., Kim, J.M., The dual space of (L(X,Y);τp) and the p-approximation property (2010) J. Funct. Anal., 259, pp. 2437-2454
  • Defant, A., Floret, K., (1993) Tensor Norms and Operator Ideals, , North Holland Publishing Co., Amsterdam
  • Delgado, J.M., Oja, E., Piñeiro, C., Serrano, E., The p-approximation property in terms of density of finite rank operators (2009) J. Math. Anal. Appl., 354, pp. 159-164
  • Delgado, J.M., Piñeiro, C., P-Convergent sequences and Banach spaces in which p-compact sets are q-compact (2011) Proc. Amer. Math. Soc., 139 (3), pp. 957-967
  • Delgado, J.M., Piñeiro, C., Serrano, E., Density of finite rank operators in the Banach space of p-compact operators (2010) J. Math. Anal. Appl., 370, pp. 498-505
  • Delgado, J.M., Piñeiro, C., Serrano, E., Operators whose adjoints are quasi p-nuclear (2010) Studia Math., 197 (3), pp. 291-304
  • Diestel, J., Fourie, J.H., Swart, J., (2008) The Metric Theory of Tensor Products, , American Mathematical Society, Providence, RI, Grothendieck's résumé revisited
  • Diestel, J., Jarchow, H., Tonge, A., Absolutely Summing Operators (1995) Cambridge Stud. Adv. Math., 43. , Cambridge University Press, Cambridge
  • Enflo, P., A counterexample to the approximation problem in Banach spaces (1973) Acta Math., 130 (1), pp. 309-317
  • Galicer, D., Lassalle, S., Turco, P., The ideal of p-compact operators: a tensor product approach (2012) Studia Math., 211 (3), pp. 269-286
  • Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires (1955) Mem. Amer. Math. Soc., 1955 (16), p. 140
  • Lassalle, S., Turco, P., On p-compact mappings and the p-approximation property (2012) J. Math. Anal. Appl., 389 (2), pp. 1204-1221
  • Lima, A., Lima, V., Oja, E., Bounded approximation properties via integral and nuclear operators (2010) Proc. Amer. Math. Soc., 138, pp. 287-297
  • Lindenstrauss, J., Tzafriri, L., (1977) Classical Banach Spaces I, vol. 92, , Springer-Verlag, Berlin, New York
  • Lissitsin, A., Mikkor, K., Oja, E., Approximation properties defined by spaces of operators and approximability in operator topologies (2008) Illinois J. Math., 52 (2), pp. 563-582
  • Oertel, F., Compositions of operator ideals and their regular hulls (1995) Acta Univ. Carolin. Math. Phys., 36 (2), pp. 69-72
  • Oja, E., On bounded approximation properties of Banach spaces (2010) Banach Center Publ., 91, pp. 219-231. , Polish Acad. Sci. Inst. Math., Warsaw, Banach Algebras, 2009
  • Oja, E., Inner and outer inequalities with applications to approximation properties (2011) Trans. Amer. Math. Soc., 363, pp. 5827-5846
  • Oja, E., A remark on the approximation of p-compact operators by finite-rank operators (2012) J. Math. Anal. Appl., 387 (2), pp. 949-952
  • Oja, E., Grothendieck's nuclear operator theorem revisited with an application to p-null sequences (2012) J. Funct. Anal., 263 (9), pp. 2876-2892
  • Pietsch, A., (1980) Operator Ideals, , North Holland Publishing Co., Amsterdam, New York, Oxford
  • Pietsch, A., The ideal of p-compact operators and its maximal hull (2013) Proc. Amer. Math. Soc., , in press
  • Ryan, R., (2002) Introduction to Tensor Products on Banach Spaces, , Springer-Verlag, London
  • Schwartz, L., Produits tensoriels topologiques d'espaces vectoriels topologiques, Exp. 14, Séminaire 1953/54, Inst. Henri Poincaré; Sinha, D.P., Karn, A.K., Compact operators whose adjoints factor through subspaces of ℓp (2002) Studia Math., 150, pp. 17-33
  • Sinha, D.P., Karn, A.K., Compact operators which factor through subspaces of ℓp (2008) Math. Nachr., 281, pp. 412-423

Citas:

---------- APA ----------
Lassalle, S. & Turco, P. (2013) . The Banach ideal of A-compact operators and related approximation properties. Journal of Functional Analysis, 265(10), 2452-2464.
http://dx.doi.org/10.1016/j.jfa.2013.07.001
---------- CHICAGO ----------
Lassalle, S., Turco, P. "The Banach ideal of A-compact operators and related approximation properties" . Journal of Functional Analysis 265, no. 10 (2013) : 2452-2464.
http://dx.doi.org/10.1016/j.jfa.2013.07.001
---------- MLA ----------
Lassalle, S., Turco, P. "The Banach ideal of A-compact operators and related approximation properties" . Journal of Functional Analysis, vol. 265, no. 10, 2013, pp. 2452-2464.
http://dx.doi.org/10.1016/j.jfa.2013.07.001
---------- VANCOUVER ----------
Lassalle, S., Turco, P. The Banach ideal of A-compact operators and related approximation properties. J. Funct. Anal. 2013;265(10):2452-2464.
http://dx.doi.org/10.1016/j.jfa.2013.07.001