  A thick-restart Lanczos type algorithm is proposed for Hermitian
$J$-symmetric matrices. Since Hermitian $J$-symmetric matrices possess doubly
degenerate spectra or doubly multiple eigenvalues with a simple relation
between the degenerate eigenvectors, we can improve the convergence of the
Lanczos algorithm by restricting the search space of the Krylov subspace to
that spanned by one of each pair of the degenerate eigenvector pairs. We show
that the Lanczos iteration is compatible with the $J$-symmetry, so that the
subspace can be split into two subspaces that are orthogonal to each other. The
proposed algorithm searches for eigenvectors in one of the two subspaces
without the multiplicity. The other eigenvectors paired to them can be easily
reconstructed with the simple relation from the $J$-symmetry. We test our
algorithm on randomly generated small dense matrices and a sparse large matrix
originating from a quantum field theory.
