On symplectic self-adjointness of Hamiltonian operator matrices

TL;DR

This paper investigates the symplectic self-adjointness of Hamiltonian operator matrices, providing necessary and sufficient conditions for different domain cases, with applications in symplectic elasticity.

Contribution

It introduces new criteria for symplectic self-adjointness of Hamiltonian operator matrices using Frobenius-Schur factorizations and perturbation methods.

Findings

Necessary and sufficient conditions for diagonal domain cases

Necessary and sufficient conditions for off-diagonal domain cases

Application to a problem in symplectic elasticity

Abstract

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.