The a Priori Tan Θ Theorem for Eigenvectors

Let A be a self-adjoint operator on a Hilbert space H. Assume that the spectrum of A consists of two disjoint components σ0 and σ1 such that the convex hull of the set σ0 does not intersect the set σ1 . Let V be a bounded self-adjoint operator on H offdiagonal with respect to the orthogonal decomposition H = H0 ⊕H1 where H0 and H1 are the spectral subspaces of A associated with the spectral sets σ0 and σ1, respectively. It is known that if ‖V‖< √ 2d where d = dist(σ0,σ1) > 0 then the perturbation V does not close the gaps between σ0 and σ1 . Assuming that f is an eigenvector of the perturbed operator A +V associated with its eigenvalue in the interval (min(σ0)− d,max(σ0)+ d) we prove that under the condition ‖V‖ < √ 2d the (acute) angle θ between f and the orthogonal projection of f onto H0 satisfies the bound tanθ ≤ ‖V‖ d and this bound is sharp.