Non-complemented spaces of operators, vector measures, and \(c_0\) (Q2909640)

The problem whether the compact operators can be a complemented subspace of the space of bounded operators was solved by Argyros and Haydon when they solved the scalar-plus-compact problem. However, spaces of operators have a clear tendency of not being complemented subspaces in \(L(X,Y)\). In the paper under review the authors, in Section 2, prove a vector measure approach to obtain the conclusion that \(K(X,Y)\) or \(W(X,Y)\) is not complemented in \(L(X,Y)\) under various conditions. This approach turns out to be a unified way to many of the known results of this type, as well as to a couple of improvements.NEWLINENEWLINEIn particular, the authors obtain the important result of Emmanuele and John that \(K(X,Y)\) is never complemented in \(L(X,Y)\) if \(K(X,Y)\) contains a copy of \(c_0\). Following this line, the authors show in Section 3 that if \(X\) contains \(\ell_1\) and there exists \(p\geq 2\) with a non-compact operator from \(\ell_p\) into \(Y\), then there has to be a copy of \(c_0\) inside \(K(X,Y)\).