The Banach ideal of A-compact operators and related approximation properties

Abstract

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on KA, the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on KA. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with KA instead of A. We exemplify the power of the Carl–Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.