On the Index of Conditional Stability of Stable Invariant Lagrangian Subspaces

Abstract

Given a nondegenerate sesquilinear inner product on a finite dimensional complex vector space, or a nondegenerate symmetric or skewsymmetric inner product on finite dimensional real vector space, subspaces that are simultaneously Lagrangian and invariant for a selfadjoint or a skewadjoint matrix with respect to the inner product are considered. The rate of conditional stability of such subspaces is studied, under small perturbations of both the inner product and the matrix. The concept of conditional stability (in contrast with unconditional stability) presupposes that one considers only those perturbed matrix and inner product for which the existence of invariant Lagrangian subspaces can be guaranteed a priori. Open problems regarding the index (= exact rate) of conditional stability are stated. Several inaccurate statements in the authors' previous works concerning the index are made precise. Finally, an application is given to conditional stability of hermitian solutions of continuous type algebraic Riccati equations.