Remarks on Spaces of Compact Operators between Reflexive Banach Spaces

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.