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

AbstractLet 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 λ0 (≠ λ̄0), we investigate the corresponding information at λ̄0 and, in particular, the unique pairing of Jordan bases for the root subspaces at λ0 and λ̄0