Notes on the subspace perturbation problem for off-diagonal perturbations

TL;DR

This paper investigates how spectral subspaces of self-adjoint operators change under off-diagonal perturbations, providing improved bounds and adapting optimization methods to this specific case.

Contribution

It adapts an optimization approach to off-diagonal perturbations, deriving stronger bounds on subspace variations compared to previous results.

Findings

Derived a new upper bound for subspace variation under off-diagonal perturbations

Showed that the optimization problem cannot be simplified to finite dimensions in this case

Provided a stronger rotation bound than previously known

Abstract

The variation of spectral subspaces for linear self-adjoint operators under an additive bounded off-diagonal perturbation is studied. To this end, the optimization approach for general perturbations in [J. Anal. Math., to appear; arXiv:1310.4360 (2013)] is adapted. It is shown that, in contrast to the case of general perturbations, the corresponding optimization problem can not be reduced to a finite-dimensional problem. A suitable choice of the involved parameters provides an upper bound for the solution of the optimization problem. In particular, this yields a rotation bound on the subspaces that is stronger than the previously known one from [J. Reine Angew. Math. (2013), DOI:10.1515/crelle-2013-0099].