The a priori tanθ theorem for eigenvectors

Let $A$ be a self-adjoint operator on a Hilbert space $\fH$. Assume that the spectrum of $A$ consists of two disjoint components $\sigma_0$ and $\sigma_1$ such that the convex hull of the set $\sigma_0$ does not intersect the set $\sigma_1$. Let $V$ be a bounded self-adjoint operator on $\fH$ off-diagonal with respect to the orthogonal decomposition $\fH=\fH_0\oplus\fH_1$ where $\fH_0$ and $\fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $\sigma_0$ and $\sigma_1$, respectively. It is known that if $\|V\|<\sqrt{2}d$ where $d=\dist(\sigma_0,\sigma_1)>0$ then the perturbation $V$ does not close the gaps between $\sigma_0$ and $\sigma_1$. Assuming that $f$ is an eigenvector of the perturbed operator $A+V$ associated with its eigenvalue in the interval $(\min(\sigma_0)-d,\max(\sigma_0)+d)$ we prove that under the condition $\|V\|<\sqrt{2}d$ the (acute) angle $\theta$ between $f$ and the orthogonal projection of $f$ onto $\fH_0$ satisfies the bound $\tan\theta\leq\frac{\|V\|}{d}$ and this bound is sharp.