Semidefinite perturbations in the subspace perturbation problem

TL;DR

This paper investigates how spectral subspaces of self-adjoint operators change under semidefinite perturbations, providing sharp estimates and adapting classical theorems to this specific case.

Contribution

It introduces a variant of the Davis-Kahan theorem tailored for semidefinite perturbations and derives precise bounds on spectral projection differences.

Findings

Sharp estimates on spectral projection differences under semidefinite perturbations

Adaptation of Davis-Kahan $ ext{sin}2 heta$ theorem to this setting

Extension of optimization approaches for spectral perturbation analysis

Abstract

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 from [SIAM J. Numer. Anal. 7 (1970), 1--46] 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 the optimization approach for general perturbations in [J. Anal. Math. 135 (2018), 313--343] to the present situation.