Perturbation of spectral subspaces and solution of linear operator equations

Abstract

Let A and B be normal operators on a Hilbert space. Let K_A and K_B be subsets of the complex plane, at distance at least δ from each other; let E be the spectral projector for A belonging to K_A, and let F be the spectral projector for B belonging to K_B. Our main results are estimates of the form δ||EF||< c||E(A − B)F||; in some special situations, the constant c is as low as 1. As an application, we prove, for an absolute constant d, that if the space is finite-dimensional and if A and B are normal with ||;A-B||≤ε/d, then the spectrum of B can be obtained from that of A (multiplicities counted) by moving each eigenvalue by at most ϵ. Our main results have equivalent formulations as statements about the operator equation AQ - QB = S. Let A and B be normal operators on perhaps different Hilbert spaces. Assume σ(A)K_A and σ(B) K_B, where K_A, K_B, and δ are as before. Then we give estimates of the forms δ||Q||⩽c||AQ − QB||.