Metric properties of the set of orthogonal projections and their applications to operator perturbation theory

TL;DR

This paper establishes the geometric structure of the set of orthogonal projections on a Hilbert space and applies this to derive sharper estimates for the variation of spectral subspaces under operator perturbations.

Contribution

It proves the set of orthogonal projections is a $rac{ ext{ extpi}}{2}$-geodesic metric space and improves bounds on spectral projection differences for perturbed operators.

Findings

The set of orthogonal projections is $rac{ ext{ extpi}}{2}$-geodesic.

New sharper estimate on the norm difference of spectral projections.

Improves upon recent results by Kostrykin, Makarov, and Motovilov.

Abstract

We prove that the set of orthogonal projections on a Hilbert space equipped with the length metric is $\frac\pi2$-geodesic. As an application, we consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators and obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unpertubed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476] are sharpened.