Abstract
The notion of p-compact sets arises naturally from Grothendieckʼs 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 κp). 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 κp-approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010).
Highlights
In the theory of Banach spaces, three concepts appear systematically related since the foundational articles by Grothendieck [21] and Schwartz [29]
We show that a Banach space E has the p-approximation property if and only if p-compact homogeneous polynomials with range on E can be uniformly approximated by finite rank polynomials
The final section is dedicated to the p-compact holomorphic mappings within the framework of holomorphy types, concept introduced by Nachbin [26,27]
Summary
In the theory of Banach spaces (or more precisely, of infinite dimensional locally convex spaces), three concepts appear systematically related since the foundational articles by Grothendieck [21] and Schwartz [29]. A Banach space E has the approximation property whenever the identity map can be uniformly approximated by finite rank operators on compact sets. If E ⊗ E, the subspace of finite rank operators, is dense in Lc(E; E), the space of continuous linear operators considered with the uniform convergence on compact sets Another classical reformulation states that E has the approximation property if F ⊗ E is dense in K(F ; E), the space of compact operators, for all Banach spaces F. The final section is dedicated to the p-compact holomorphic mappings within the framework of holomorphy types, concept introduced by Nachbin [26,27] This allows us to inspect the κp -approximation property introduced, in [13], in the spirit of [5, Theorem 4.1]
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