On p-compact mappings and the p-approximation property

• Autores: Lassalle, S.; Turco, P.
• Año de publicación: 2012
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Math. Anal. Appl. 2012;389(2):1204-1221
• Materia: Approximation properties; Holomorphic mappings; P-Compact sets
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

The notion of p-compact sets arises naturally from Grothendieck's characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm κ p). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ε-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the κ p-approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010). © 2012 Elsevier Inc.
Fil:Lassalle, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.