Abstract
A classification of weakly compact multiplication operators on L(L-p), 1 <p <infinity, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of l(p)-strictly singular operators, and we also investigate the structure of general l(p)-strictly singular operators on L-p. The main result is that if an operator T on L-p, 1 <p <2, is l(p)-strictly singular and T-vertical bar X is an isomorphism for some subspace X of L-p, then X embeds into L-r for all r <2, but X need not be isomorphic to a Hilbert space. It is also shown that if T is convolution by a biased coin on L-p of the Cantor group, 1