Sharpening the norm bound in the subspace perturbation theory

Abstract

Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma} is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0 where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral projection of A associated with the spectral set {\sigma} and let Q be the spectral projection of L corresponding to the closed ||V||-neighborhood of {\sigma}. We prove a bound of the form arcsin(||P-Q||)\leq M(||V||/d), M: [0,1/2)-->R^+, that is essentially stronger than the previously known estimates for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||<1 and, thus, that the spectral subspaces Ran(P) and Ran(Q) are in the acute-angle case whenever ||V||<cd with c=0.454169... (the precise expression for c is also given). Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic \sin2\theta estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.