Angles between infinite dimensional subspaces with applications to the Rayleigh–Ritz and alternating projectors methods

Abstract

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961/1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh–Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation.