The A Priori Tan $\Theta$ Theorem for Eigenvectors

Let $A$ be a self-adjoint operator on a Hilbert space $\mathfrak{H}$. 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 $\mathfrak{H}$ off-diagonal with respect to the orthogonal decomposition $\mathfrak{H}=\mathfrak{H}_0\oplus\mathfrak{H}_1$, where $\mathfrak{H}_0$ and $\mathfrak{H}_1$ are the spectral subspaces of $A$ associated with the spectral sets $\sigma_0$ and $\sigma_1$, respectively. It is known that if $\|V\|0$, 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 $(\mathrm{min}(\sigma_0)-d,\mathrm{max}(\sigma_0)+d)$, we prove that under the condition $\|V\|