A Note on Generalized Approximation Property

Antara Bhar,Manjul Gupta

doi:10.1155/2013/325141

Antara Bhar, Manjul Gupta

Open Access

https://doi.org/10.1155/2013/325141

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Abstract

We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for () has also been characterized.

Highlights

It is well known that the identity on an infinite dimensional Banach space is never compact, though it may be approximated by finite rank operators in the pointwise convergence topology, for instance, in the case when X has a Schauder base
If the identity on X is approximated uniformly on compact sets by finite rank operators, it leads to the notion of approximation property of X, studied systematically by Grothendieck [1] in 1955
In our recent work, using the duality theory of sequence spaces, we considered the notion of λ-compact sets corresponding to a suitably restricted sequence space λ and studied λ-compact operators, specially their relationships with λsumming, λ-nuclear, and quasi-λ-nuclear which were earlier considered by Ramanujan [6] in 1970

Summary

Introduction

It is well known that the identity on an infinite dimensional Banach space is never compact, though it may be approximated by finite rank operators in the pointwise convergence topology, for instance, in the case when X has a Schauder base. If the identity on X is approximated uniformly on compact sets by finite rank operators, it leads to the notion of approximation property of X, studied systematically by Grothendieck [1] in 1955. There are many reformulations of this property, all of them involve either subspaces or ideals of operators, for example, class of finite rank operators, compact operators, and so forth. This property includes, as particular cases, approximation property, papproximation property, compact approximation property, and subspace approximation property (cf [2,3,4,5])

Preliminaries

L-Subset λ-Approximation Property

L-Subset p-Approximation Property