  We introduce and study the following modified version of the Invariant
Subspace Problem: whether every operator T on a Banach space has an almost
invariant half-space, that is, a subspace Y of infinite dimension and infinite
codimension such that Y is of finite codimension in T(Y). We solve this problem
in the affirmative for a large class of operators which includes quasinilpotent
weighted shift operators on l_p (1 \le p < \infty) or c_0.
