This is an announcement for the paper "On p-compact mappings and p-approximation" by Silvia Lassalle and Pablo Turco.

Abstract: 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 a ideal with its ideal norm $\kappa_p$. This paper examines the interaction between the $p$-approximation property and the space of holomorphic functions. Here, the $p$-compact analytic functions play a crucial role. In order to understand this type of functions we define a $p$-compact radius of convergence which allow 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, defined by Schwartz, to characterize the $p$-approximation property of a Banach space in terms of $p$-compact homogeneous polynomials and also in terms of $p$-compact holomorphic functions with range on the space. Finally, we show that $p$-compact holomorphic functions fit in the framework of holomorphy types which allows us to inspect the $\kappa_p$-approximation property. Along these notes we solve several questions posed by Aron, Maestre and Rueda.

Archive classification: math.FA

Mathematics Subject Classification: 46G20, 46B28

Remarks: 31 pages

Submitted from: pabloaturco@yahoo.com.ar

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1107.1670

or

http://arXiv.org/abs/1107.1670