, Volume 26, Issue 2, pp 445-469

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.