naa a22 4500 510760856 CHVBK 20180411083138.0 cr unu---uuuuu 180411e20130701xx s 000 0 eng 10.1007/s13163-012-0107-x doi (NATIONALLICENCE)springer-10.1007/s13163-012-0107-x The structure of compact linear operators in Banach spaces [Elektronische Daten] [D. Edmunds, W. Evans, D. Harris] 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. Universidad Complutense de Madrid, 2012 Compact linear operators nationallicence Strictly convex Banach spaces nationallicence Approximation property nationallicence Basis nationallicence Edmunds D. Department of Mathematics, Sussex University, BN1 9QH, Brighton, UK aut Evans W. School of Mathematics, Cardiff University, CF24 4AG, Cardiff, Wales, UK aut Harris D. School of Mathematics, Cardiff University, CF24 4AG, Cardiff, Wales, UK aut Revista Matemática Complutense Springer Milan 26/2(2013-07-01), 445-469 1139-1138 26:2<445 2013 26 13163 https://doi.org/10.1007/s13163-012-0107-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s13163-012-0107-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Edmunds D. Department of Mathematics, Sussex University, BN1 9QH, Brighton, UK aut NATIONALLICENCE 700 1- Evans W. School of Mathematics, Cardiff University, CF24 4AG, Cardiff, Wales, UK aut NATIONALLICENCE 700 1- Harris D. School of Mathematics, Cardiff University, CF24 4AG, Cardiff, Wales, UK aut NATIONALLICENCE 773 0- Revista Matemática Complutense Springer Milan 26/2(2013-07-01), 445-469 1139-1138 26:2<445 2013 26 13163 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer