Perturbation of Invariant Subspaces

Abstract

The questions of how perturbation of an operator affects its invariant subspaces have raised a substantial amount of interest, especially in the Ž w x . finite dimensions see, e.g., 5, 10]12 and the references given there . It turns out that linear perturbation of the operator gives rise to an analytic perturbation of the projection on its invariant subspace under some reasonable conditions. Although the question of estimating the norms of the terms of the Taylor series’ expansion of this projection-valued function does have some applications in numerical analysis, it seems to be studied w x less frequently. For instance, in 12 the matrix algebra approach combined with complex analysis in several variables was used to get existence and analyticity of the perturbed projection. The terms were obtained as well, using specially developed formulas for their direct computation. However, we find it hard to see that these estimates are good enough to ensure the convergence of this series, let alone to give an estimate of its radius of convergence. In this article we propose a different approach to the subject which achieves the two goals at one stroke and, we believe, with better results. Namely, we study the problem in terms of operator-valued analytic functions, thus reducing the problem to solving an algebraic Riccati-type equation. In order to solve it we extend in Sections 2 and 3 somewhat the Ž usual methods to our case see the comments about the meaning of . ‘‘usual’’ in the last paragraph of the introduction . The main result, given in Section 4, is the estimates of the radius of convergence and of the terms. The estimates are clearly good enough to yield an estimated radius of