The local determination of Jordan bases for H-self-adjoint operators

Abstract

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space X . A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A ∗ H. Let σ( A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A ∗ and λ ∈ σ( A) implies λ̄ ∈ σ ( A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ 0 (≠ λ̄ 0), we investigate the corresponding information at λ̄ 0 and, in particular, the unique pairing of Jordan bases for the root subspaces at λ 0 and λ̄ 0.