Proximinality of certain spaces of compact operators (Q2712537)

A closed subspace \(J\) of a Banach space \(X\) is proximinal in \(X\) if for every \(x\in X_J\) there exists \(j_0\in J\) such that \(|x-j_0|= \inf \{|x-j|: j\in J\}\). If \(X\) and \(Y\) are two Banach spaces, then \(K (X,Y)\) denotes the space of compact linear operators from \(X\) to \(Y\), and \(L(X, Y)\) denotes the space of bounded linear operators from \(X\) to \(Y\); \(K(X):K(X,X) \), \(L(X):L(X,X)\). The paper is devoted to an investigation of the proximinality of spaces of compact operators. First, using a theorem of \textit{G. Godini} [Rev. Roum. Math. Pures Appl. 18, 901-906 (1973; Zbl 0259.46020)] the authors easily obtain a very useful general result about a proximinality: Suppose \(Y\) is a closed subspace of a Banach space \(Z\). If \(Z_1\) is a proximinal subspace of \(Z\) and \(Y_1\) is a closed subspace of \(Y\) with \(Y_1\subseteq Z_1\), then \(Y_1\) is proximinal in \(Y\).NEWLINENEWLINENEWLINEAs a corollary, the authors find a larger class of spaces of compact operators which are proximinal in the corresponding spaces of bounded linear operators: Suppose \(X\) and \(Y\) are are Banach spaces for which \(K(X,Y)\) is proximinal in \(L(X,Y)\). If \(Z\) is a closed subspace in \(Y\), then \(K(X,Z)\) is proximinal in \(L(X,Z)\). In particular, the authors obtain the proximinality of \(K(X)\) in \(L(X)\) for the case when \(X\) is any closed subspace of \(I_p\) \((1<p<\infty)\). The second main result is the following: If \(X\) is a Banach space with an unconditional shrinking basis, then \(K(X,c_0)\) is proximinal in\((X,I_\infty)\). The authors use throughout the notion of an \(M\)-ideal introduced by \textit{E. M. Alfsen} and \textit{E. G. Effros} [Ann. of Math. (2) 96, 98-128 (1972; Zbl 0248.46019)] and its characterization by \(n\)-ball properties of closed balls to prove the results. The article contains also an extensive bibliography of results obtained in this area.