The Compact Approximation Property for Weighted Spaces of Holomorphic Mappings

Abstract

In this paper, we examine the compact approximation property for the weighted spaces of holomorphic functions. We show that a Banach space E has the compact approximation property if and only if the predual \(\mathcal {G}_v(U)\) of the space \(H_v(U)\) consisting of all holomorphic mappings \(f:U\rightarrow \mathbb {C}\) (complex plane) with \(\sup \limits _{x\in U}v(x)\Vert f(x)\Vert <\infty \) has the compact approximation property, where v is a radial weight defined on a balanced open subset U of E such that \(H_v(U)\) contains all the polynomials. We have also studied the compact approximation property for the weighted (LB)-space VH(E) of holomorphic mappings and its predual VG(E) for a countable decreasing family V of radial rapidly decreasing weights on E.