On an estimate in the subspace perturbation problem

The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. In the approach presented here, a constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate on the arcsine of the norm of the difference of the corresponding spectral projections. The problem is solved explicitly. This optimizes the approach by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. In particular, the resulting estimate is stronger than the one obtained there.