  Symplectic self-adjointness of Hamiltonian operator matrices is studied,
which arises in symplectic elasticity and optimal control. For the cases of
diagonal domain and off-diagonal domain, necessary and sufficient conditions
are shown. The proofs use Frobenius-Schur fractorizations of unbounded operator
matrices. Under additional assumptions, sufficient conditions based on
perturbation method are obtained. The theory is applied to a problem in
symplectic elasticity.
