  A classification of weakly compact multiplication operators on L(L_p),
$1<p<\infty$, is given. This answers a question raised by Saksman and Tylli in
1992. The classification involves the concept of $\ell_p$-strictly singular
operators, and we also investigate the structure of general $\ell_p$-strictly
singular operators on L_p. The main result is that if an operator T on L_p,
1<p<2, is $\ell_p$-strictly singular and T_{|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, $1\le p <2$, and $T_{|X}$ is an isomorphism for some reflexive
subspace X of L_p, then X is isomorphic to a Hilbert space. The case p=1
answers a question asked by Rosenthal in 1976.
