The a priori Tan Theta Theorem for spectral subspaces

Abstract

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1. It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the perturbation V does not close the gaps between s_0 and s_1 and the spectrum of the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1 grown from s_0 and s_1, respectively. Furthermore, it is known that if V satisfies the stronger bound ||V||< d then the following sharp norm estimate holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and E_L(s'_0) are the spectral projections of A and L associated with the spectral sets s_0 and s'_0, respectively. In the present work we prove that this estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which completely settles the issue.