A classification of weakly compact multiplication operators on , , is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of -strictly singular operators, and we also investigate the structure of general -strictly singular operators on . The main result is that if an operator on , , is -strictly singular and is an isomorphism for some subspace of , then embeds into for all , but need not be isomorphic to a Hilbert space. It is also shown that if is convolution by a biased coin on of the Cantor group, , and is an isomorphism for some reflexive subspace of , then is isomorphic to a Hilbert space. The case answers a question asked by Rosenthal in 1976.