Journal of Operator Theory

Volume 81, Issue 2, Spring 2019 pp. 321-333.

Semidefinite perturbations in the subspace perturbation problem

Authors: Albrecht Seelmann
Author institution:Fakultaet fuer Mathematik, Technische Universitaet Dortmund, Dortmund, D-44221, Germany

Summary: The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan $\sin2\Theta$ theorem adapted to this situation is proved. Under a certain additional geometric assumption on the separation of the spectrum of the unperturbed operator, this leads to a sharp estimate on the norm of the difference of the spectral projections associated with isolated components of the spectrum of the perturbed and unperturbed operators, respectively. Without this additional geometric assumption on the isolated components of the spectrum of the unperturbed operator, a corresponding estimate is obtained by transferring a known optimization approach for general perturbations to the present situation.

DOI: http://dx.doi.org/10.7900/jot.2018feb07.2186
Keywords: subspace perturbation problem, spectral subspaces, maximal angle between closed subspaces, semidefinite perturbations

Contents Full-Text PDF