Complemented subspaces of linear bounded operators (Q2909631)

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 classical results of Kalton and results on separably determined operator ideals with a so-called property \((\ast)\) are used to derive general theorems on the non-complementation of operator spaces in \(L(X,Y)\). It is shown that spaces of compact, weakly compact, and completely continuous operators enjoy property \((\ast)\) and so the general results which are proved imply lots of important known results on the non-complementation of these spaces in \(L(X,Y)\).