Characterizations of approximation properties defined by operator ideals in the predual of weighted Banach spaces of holomorphic functions

TL;DR

This paper investigates the approximation properties of preduals of weighted holomorphic function spaces, establishing a connection with the underlying Banach space's properties via operator ideals.

Contribution

It characterizes when the predual space has the -approximation property based on the Banach space's approximation property, linking operator ideals and weighted holomorphic functions.

Findings

Predual  of weighted holomorphic spaces has the -approximation property iff the underlying Banach space does.

Establishes a precise equivalence relating approximation properties of preduals and Banach spaces.

Connects operator ideals with approximation properties in the context of weighted holomorphic functions.

Abstract

In this article, we show that the predual $\mathcal{G}_w(U)$ of the weighted space of holomorphic functions has the $\mathcal{I}$- approximation property if and only if $E$ has the $\mathcal{I}$- approximation property, where $\mathcal{I}$ is a suitably chosen operator ideal, and $\it{w}$ is a radial weight defined on a balanced open subset U of a Banach space $E$.