Transformation and Perturbation of Subspaces of a Banach Space

Abstract

<p>An n-frame on a Banach space X is E=(E₁,...,En) where the Ej's are bounded linear operators on X such that Ej≠0, ∑ Ej=I and EjEk=δjkEk (j,k=1,2,...,n). This with the study of pairs of such n-frames. It is shown that if two n-frames are close to each other then they are similar. A particular similarity, the direct rotation comes naturally in connection with the geodesic arc connecting the two frames when the set of n-frames in regarded as a Banach manifold. For a pair of 2-frames, the direct rotation is characterized. Another similarity, the balanced transformation which realizes the equivalence of the two frames is locally characterized and its closeness to the direct rotation is investigated. These results are used to obtain an error bound on invariant subspaces under perturbation. Our study, which is based on a functional calculus approach, involves techniques and results from operator theory, perturbation theory, and differential geometry. Some of the results are relevant to numerical spectral analysis.</p>