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 \[ \mathsf {E}_A(\sigma )\quad \text {and} \quad \mathsf {E}_{A+V}\big (\{\lambda | \mathrm {dist}(\lambda , \sigma )<d/2\}\big )\] for $A$ and $A+V$ is less than one whenever either (i) $\|V\|<\frac {2}{2+\pi }d$ or (ii) $\|V\|<\frac {1}{2}d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.