Abstract
Following Sinha and Karn [9], a relatively compact subset K of a Banach space E is said to be p-compact if for some sequence (x n ) ∈ l p (E), K ⊂ {Σ n a x x n | (a n ) ∈ B ℓ′ p }. In [4], Delgado, Oja, Piñeiro, and Serrano investigated the p-approximation property, in which one only requires finite rank approximation of the identity on p-compact subsets. We investigate analogous concepts here for the case of holomorphic mappings between Banach spaces, introducing the space of p-compact holomorphic mappings (cf. [1]). A number of problems related to such holomorphic mappings are discussed.
Resumen.
Según Sinha y Karn [9], un subconjunto relativamente compacto K de un espacio de Banach E es p-compacto si para alguna sucesión (x n ) ∈ l p (E), K ⊂ {Σ n a x x n | (a n ) ∈ B ℓ′ p }. En [4], Delgado, Oja, Piñeiro, y Serrano investigaron la propiedad de p-aproximación, en la que sólo se requiere aproximaciones en conjuntos p-compactos. En este trabajo investigamos conceptos análogos para aplicaciones holomorfas entre espacios de Banach, introduciendo el espacio de aplicaciones holomorfas p-compactas (cf. [1]). También presentamos una colección de problemas relacionados con dichas aplicaciones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aron, R. M. and Schottenloher, M., (1976). Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal., 21, 7–30. DOI: 10.1016/0022-1236(76)90026-4
Barroso J. A.; Matos, M. C. and Nachbin, L., (1977). On holomorphy versus linearity in classifying locally convex spaces. In Infinite Dimensional Holomorphy and applications, M. C. Matos, ed., North-Holland Math. Studies, 12, North-Holland, 31-74. DOI: 10.1016/S0304-0208(08)72193-5
Bierstedt, K. D. and Meise, R., (1974). Bemerkungen über die Approximations-eigenschaft lokalkonvexer Funktionenräume, Math. Ann., 209, 99–107.
Delgado, J. A.; Oja, E.; Piñeiro, C. and Serrano, E., (2009). The p-approximation property in terms of density of finite rank operators, J. Math. Anal. Appl., 354, 1, 159–164. DOI: 10.016/j.jmaa.2008.12.047
Diestel, J. and Uhl, J. J. Jr., (1977). Vector measures, Mathematical Surveys andMonographs, 15, American Mathematical Society.
Dineen, S., (1981). Complex analysis in locally convex spaces, Mathematics Studies, 57, Mathematical Notes, 83, North-Holland Publishing Co., Amsterdam-New York.
Dineen, S., (1999). Complex analysis on infinite-dimensional spaces, Monographs in Mathematics, Springer-Verlag London, Ltd., London.
Schwartz, L., (1957). Théorie des distributions à valeurs vectorielles I, Ann. Inst. Fourier (Grenoble), 7, 1–141.
Sinha, D. P. and Karn, A. K., (2002). Compact operators whose adjoints factor through subspaces of l p , Studia Math., 150, 17–33. DOI: 10.4064/sm150-1-3
Sinha, D. P. and Karn, A. K., (2008). Compact operators which factor through subspaces of l p , tMath. Nachr., 281, 412–423.
Author information
Authors and Affiliations
Corresponding author
Additional information
In honor of Don Manuel Valdivia, who has always searched for and, almost always, solved problems
Submitted by J. Bonet
Rights and permissions
About this article
Cite this article
Aron, R.M., Maestre, M. & Rueda, P. p-Compact holomorphic mappings. RACSAM 104, 353–364 (2010). https://doi.org/10.5052/RACSAM.2010.22
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.5052/RACSAM.2010.22