The a priori tanθtheorem for eigenvectors

Abstract

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 $\Vert V\Vert <\sqrt{2}d$ where $d=\dist(σ_0,σ_1)>0$ 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 $\Vert V\Vert <\sqrt{2}d$ the (acute) angle $\theta$ between $f$ and the orthogonal projection of $f$ onto $\fH_0$ satisfies the bound $\tan \theta \le \frac{\Vert V\Vert }{d}$ and this bound is sharp.

References in corpus (3)

• On the existence of solutions to the operator Riccati equation and the tanΘtheorem
• Perturbation of spectra and spectral subspaces
• Some sharp norm estimates in the subspace perturbation problem