Approximation properties determined by operator ideals and approximability of homogeneous polynomials and holomorphic functions

Abstract

Given an operator ideal I, a Banach space E has the I-approximation property if the identity operator on E can be uniformly approximated on compact subsets of E by operators belonging to I. In this paper the I-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of linear operators/homogeneous polynomials, spaces of holomorphic functions and their preduals.1. Introduction. Given Banach spaces E and F , we denote by L(E; F ) the Banach space of all bounded linear operators from E to F endowed with the usual operator sup norm. The subspaces of L(E; F ) formed by all finite rank, all compact and all weakly compact operators are denoted by F(E; F ), K(E; F ) and W(E; F ), respectively. For a subset S of L(E; F ), the symbol S τc represents the closure of S with respect to the compact-open topology τ c .By id E we denote the identity operator on E.

Ideal topologies and corresponding approximation properties

“…In this section we reinforce the unifying feature of our approach to approximation properties via ideal topologies by proving that some recent results of [16,6,10] on approximation properties in projective tensor products of Banach spaces are particular instances of much more general results in the realm of ideal topologies. Remember that approximation properties and topological tensor products are closely connected since Grothendieck [30].…”

“…(c) Let I be an operator ideal. The I-approximation property of [6] coincides with the (L, I, τ c )-AP (hence with the (L, I, τ c )-WAP).…”

“…We finish the paper showing that the concept of projective ideal topology allows us to generalize the results of [6,Section 3]. We shall need the so-called factorization method to generate a multi-ideal from a given operator ideal.…”

“…The consideration of problems on the approximation by operators belonging to a given operator ideal was a question of time. Indeed, a number of approximation properties (APs) with respect to operator ideals-and other ones that are somehow related to operator ideals-have been studied in the last three decades, see, e.g., [6,11,13,15,19,20,29,34,35,37,38,39,40,41,49,50,57,58,59,62,64]. The reader is also referred to the surveys [51,52] and to the references therein.…”

“…In Section 3 we extend/generalize results from [19,14] on APs to the language of (I, J , τ )-APs. To reinforce the unifying feature of our approach, in Section 4 we introduce the notion of projective ideal topology to prove that recent results from [16,6,10] on APs in (symmetric) projective tensor products of Banach spaces are particular instances of much more general results in the context of (I, J , τ )-APs.…”