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 A and V V be bounded self-adjoint operators. Assume that the spectrum of A A consists of two disjoint parts σ \sigma and Σ \Sigma such that d = dist ( σ , Σ ) > 0 d=\text {dist}(\sigma , \Sigma )>0 . We show that the norm of the difference of the spectral projections \[ E A ( σ ) and E A + V ( { λ | d i s t ( λ , σ ) > d / 2 } ) \mathsf {E}_A(\sigma )\quad \text {and} \quad \mathsf {E}_{A+V}\big (\{\lambda \,\, | \,\, \mathrm {dist}(\lambda , \sigma )>d/2\}\big ) \] for A A and A + V A+V is less than one whenever either (i) ‖ V ‖ > 2 2 + π d \|V\|>\frac {2}{2+\pi }d or (ii) ‖ V ‖ > 1 2 d \|V\|>\frac {1}{2}d and certain assumptions on the mutual disposition of the sets σ \sigma and Σ \Sigma are satisfied.