On p-compact mappings and the p-approximation property

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:389

Article

Turco, Pablo¹ 

[1] Univ Buenos Aires, Fac Ciencias Exactas & Nat, Dept Matemat, RA-1428 Buenos Aires, DF, Argentina

关键词: p-Compact sets;    Holomorphic mappings;    Approximation properties;   

DOI  :  10.1016/j.jmaa.2011.12.058

来源: Elsevier

【 摘 要 】

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 k(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 epsilon-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 k(p)-approximation property. Our approach also allows us to solve several questions posed by Aron. Maestre and Rueda (2010). (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free