  Recently, Durkee and Reall have conjectured a criterion for linear
instability of rotating, extremal, asymptotically Minkowskian black holes in
$d\ge 4$ dimensions, such as the Myers-Perry black holes. They considered a
certain elliptic operator, $\cA$, acting on symmetric trace-free tensors
intrinsic to the horizon. Based in part on numerical evidence, they suggested
that if the lowest eigenvalue of this operator is less than the critical value
$-1/4$ ( called "effective BF-bound"), then the black hole is linearly
unstable. In this paper, we prove an extended version of their conjecture. Our
proof uses a combination of methods such as (i) the "canonical energy method"
of Hollands-Wald, (ii) algebraically special properties of the near horizon
geometries associated with the black hole, (iii) the Corvino-Schoen technique,
and (iv) semiclassical analysis. Our method of proof is also applicable to
rotating, extremal asymptotically Anti-deSitter black holes. In that case, we
find additional instabilities for ultra-spinning black holes. Although we
explicitly discuss in this paper only extremal black holes, we argue that our
results can be generalized to {\em near} extremal black holes.
