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The a priori tanθtheorem 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 $σ_0$ and $σ_1$ such that the convex hull of the set $σ_0$ does not intersect the set $σ_1$. 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 $σ_0$ and $σ_1$, respectively. It is known that if $\|V\|<\sqrt{2}d$ where $d=\dist(σ_0,σ_1)>0$ then the perturbation $V$ does not close the gaps between $σ_0$ and $σ_1$. Assuming that $f$ is an eigenvector of the perturbed operator $A+V$ associated with its eigenvalue in the interval $(\min(σ_0)-d,\max(σ_0)+d)$ we prove that under the condition $\|V\|<\sqrt{2}d$ the (acute) angle $θ$ between $f$ and the orthogonal projection of $f$ onto $\fH_0$ satisfies the bound $\tanθ\leq\frac{\|V\|}{d}$ and this bound is sharp.