Error Bounds for Approximate Invariant Subspaces of Closed Linear Operators

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