Operators ideals and approximation properties

2012/11/30 by Lassalle, Silvia, Turco, Pablo
#46B20 #46B28 #47B07 #47L20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1211.7366

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 ε-product.

Related

• Compactness and an approximation property related to an operator ideal
• A general abstract approach to approximation properties in Banach spaces
• Approximation properties determined by operator ideals
• The p-Operator Approximation Property
• Weaker relatives of the bounded approximation property for a Banach\n operator ideal