Error Bounds for Approximate Invariant Subspaces of Closed Linear Operators

Let A be a closed linear operator on a separable Hilbert space H whose domain is dense in H Let X be a subspace of H contained in the domain of A and let Y be its orthogonal complement. Let B and C be the compressions of A to Z and Y respectively, let G = Y^ * AX, where X and Y are the injections of X and Y into H. It is shown that if B and C have disjoint spectra and \| G \| is sufficiently small, then there is an invariant subspace X' of A near X. Bounds for the distance between X' and X are given, and the spectrum of A is related to the spectra of B and C. In the development a measure of the separation of the spectra of B and C which is insensitive to small perturbations in B and C is introduced and analyzed.