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.

Cited by in corpus (12)

• On the existence of solutions to the operator Riccati equation and the tanΘtheorem
• Perturbation of spectra and spectral subspaces
• Notes on the sin2Θ theorem
• Sharpening the norm bound in the subspace perturbation theory
• Bounds on variation of the spectrum and spectral subspaces of a few-body Hamiltonian
• On invariant graph subspaces
• Notes on the subspace perturbation problem for off-diagonal perturbations
• On an estimate in the subspace perturbation problem
• Metric properties of the set of orthogonal projections and their applications to operator perturbation theory
• On applying the subspace perturbation theory to few-body Hamiltonians
• Semidefinite perturbations in the subspace perturbation problem
• On an integral formula for Fredholm determinants related to pairs of spectral projections