  We show that any bounded operator $T$ on a separable, reflexive,
infinite-dimensional Banach space $X$ admits a rank one perturbation which has
an invariant subspace of infinite dimension and codimension. In the
non-reflexive spaces, we show that the same is true for operators which have
non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our
methods produce perturbations that are also small in norm, improving on an old
result of Brown and Pearcy.
