  We show that for any bounded operator $T$ acting on an infinite dimensional
Banach space there exists an operator $F$ of rank at most one such that $T+F$
has an invariant subspace of infinite dimension and codimension. We also show
that whenever the boundary of the spectrum of $T$ or $T^*$ does not consist
entirely of eigenvalues, we can find such rank one perturbations that have
arbitrarily small norm. When this spectral condition is not satisfied, we can
still find suitable finite rank perturbations of arbitrarily small norm, but
not necessarily of rank one.
