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Remarks on Spaces of Compact Operators between Reflexive Banach Spaces

  • G. GodefroyEmail author
Conference paper
  • 448 Downloads
Part of the Operator Theory: Advances and Applications book series (OT, volume 240)

Abstract

We observe that if X and Y are two reflexive separable spaces such that the canonical map \(J\;:\;X\hat{\bigotimes}Y\;\rightarrow\;X\check{\bigotimes}Y\) is injective, then every compact operator from \(X\;\mathrm{to}\;Y^{\ast}\) is in the norm closure of finite rank operators, and every bounded operator \(T\;\in\;L(X,Y^{\ast})\) is uniform limit on compact sets of a sequence \((R_{n})\) of finite rank operators such that \(\parallel R_{n}\parallel\;\leq\;\parallel T\parallel\). This would apply in particular to the case \(X\;=\;Y\), i.e., to a reflexive Pisier space if such a space exists. We show that if \(Z\;\subset\;L(X)\)is a subspace which strictly contains the space \(K(X)\) of compact operators on a reflexive Banach space X, then \(K(X)\) is not 1-complemented in Z, and it is locally 1-complemented in Z exactly when Z is contained in the closure of \(K(X)\) with respect to the uniform convergence on compact subsets of X. Several consequences are spelled out.

Keywords

Spaces of compact operators approximation properties smooth norms 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheCNRS-Université Paris 6Paris Cedex 05France

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