The structure of compact linear operators in Banach spaces

Edmunds, D. E., Evans, William Desmond and Harris, D. J. 2013. The structure of compact linear operators in Banach spaces. Revista Mathematica Complutense 26 (2) , pp. 445-469. 10.1007/s13163-012-0107-x

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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 xn∈X, projections Pn of X onto subspaces Xn which are such that ∩Xn=kerT, and linear projections Sn satisfying Snx=∑n−1j=1ξj(x)xj, where the coefficients ξj(x) are given explicitly. If kerT={0} and the condition (A):sup∥Sn∥<∞ is satisfied, the representation reduces to an analogue of the Schmidt representation of T when X and Y are Hilbert spaces, and also (xn) 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.

Item Type:	Article
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer
Last Modified:	04 Jun 2017 08:57
URI:	https://orca.cardiff.ac.uk/id/eprint/88266

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