On a Subspace Perturbation Problem

Abstract

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let $A$ and $V$ be bounded self-adjoint operators. Assume that the spectrum of $A$ consists of two disjoint parts $\sigma$ and $\Sigma$ such that $d=\text{dist}(\sigma ,\Sigma )>0$. We show that the norm of the difference of the spectral projections $\EE_A(σ)$ and $\EE_{A+V}\big (\{λ| \dist(λ, σ)$ $<d/2\})$ for $A$ and $A+V$ is less then one whenever either (i) $\Vert V\Vert <\frac{2}{2+\pi }d$ or (ii) $\Vert V\Vert <1/2d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.