  In this paper we consider the possible existence of unstable axisymmetric
modes in Kerr space times, resulting from exponentially growing solutions of
the Teukolsky equation. We describe a transformation that casts the radial
equation that results upon separation of variables in the Teukolsky equation,
in the form of a Schr\"odinger equation, and combine the properties of the
solutions of this equations with some recent results on the asymptotic
behaviour of spin weighted spheroidal harmonics to prove the existence of an
infinite family of unstable modes. Thus we prove that the stationary region
beyond a Kerr black hole inner horizon is unstable under gravitational linear
perturbations. We also prove that Kerr space-time with angular momentum larger
than its square mass, which has a naked singularity, is unstable.
