, Volume 26, Issue 2, pp 445-469
Date: 16 Oct 2012

The structure of compact linear operators in Banach spaces

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Abstract

In Edmunds et al. [J Lond Math Soc 78(2):65–84, 2008], a representation of a compact linear operator \(T\) acting between reflexive Banach spaces \(X\) and \(Y\) with strictly convex duals was established in terms of elements \(x_n \in X,\) projections \(P_n\) of \(X\) onto subspaces \(X_n\) which are such that \( \cap X_n = \mathrm {ker}{\rm T},\) and linear projections \(S_n\) satisfying \(S_n x = \sum _{j=1}^{n-1} \xi _j(x) x_j,\) where the coefficients \(\xi _j(x)\) are given explicitly. If \(\mathrm {ker}{\rm T} = \{0\}\) and the condition $$\begin{aligned} (A): \quad sup \Vert S_n \Vert < \infty \end{aligned}$$ is satisfied, the representation reduces to an analogue of the Schmidt representation of \(T\) when \(X\) and \(Y\) are Hilbert spaces, and also \((x_n)\) is a Schauder basis of \(X\) ; thus condition (A) can not be satisfied if \(X\) does not have the approximation property. In this paper we investigate circumstances in which (A) does or does not hold, and analyse the implications.