  We study the interpolation and extrapolation properties of strictly singular
operators between different $L_p$ spaces. To this end, the structure of
strictly singular non-compact operators between $L_p-L_q$ spaces is analyzed.
Among other things, we clarify the relation between strict singularity and the
$L$-characteristic set of an operator. In particular, Krasnoselskii's
interpolation theorem for compact operators is extended to the class of
strictly singular operators.
