Geodesic
Projections from $L(X,Y)$ onto $K(X,Y)$
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 765-771.

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^*$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^*$ has the approximation property). Suppose that $L(X,Y)\neq K(X,Y)$ and let $1\lambda2$. Then $X$ (respectively $Y$) can be equivalently renormed so that any projection $P$ of $L(X,Y)$ onto $K(X,Y)$ has the sup-norm greater or equal to $\lambda $.
Classification : 46B28
Keywords: compact operator; approximation property; reflexive Banach space; projection; separability 
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John, Kamil. Projections from $L(X,Y)$ onto $K(X,Y)$. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 765-771. http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a9/