Operators ideals and approximation properties

TL;DR

This paper introduces a unified framework using Banach operator ideals to study approximation properties and operator ideals, providing new insights and generalizations of classical theorems in functional analysis.

Contribution

It systematically extends classical approximation properties via the notion of extit{A}-compact sets and develops a norm on extit{A}-compact operators, broadening the understanding of operator ideals.

Findings

extit{K}_ extit{A} is a dual, regular ideal with a characterized maximal hull.

Introduces a norm on extit{A}-compact operators to measure set size.

Generalizes Schwartz theorem using a revisited extit{ε}-product.

Abstract

We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the $\A$-approximation property if the identity map is uniformly approximable on $\A$-compact sets by finite rank operators. The Grothendieck's classic approximation property is the $\K$-approximation property for $\K$ the ideal of compact operators and the $p$-approximation property is obtained as the $\mathcal N^p$-approximation property for $\mathcal N^p$ the ideal of right $p$-nuclear operators. We introduce a way to measure the size of $\A$-compact sets and use it to give a norm on $\K_\A$, the ideal of $\A$-compact operators. Most of our results concerning the operator Banach ideal $\K_\A$ are obtained for right-accessible ideals $\A$. For instance, we prove that $\K_\A$ is a dual ideal, it is regular and we characterize its maximal hull. A strong concept of approximation property, which makes use of the norm defined on $\K_\A$, is also addressed. Finally, we obtain a generalization of Schwartz theorem with a revisited $\epsilon$-product.