Schur multiplier projections on the von Neumann-Schatten classes

For 1 < p < infinity let C-p denote the usual von Neumann-Schatten ideal of compact operators on l(2). The standard basis of C-p is a conditional one and so it is of interest to be able to identify the sets of coordinates for which the corresponding projection is bounded. In this paper we survey and extend the known classes of bounded projections of this type. In particular we show that some recent results from spectral theory allow one to prove boundedness of a projection by checking simple geometric conditions on the associated set of coordinates.