Topics In Theory Of Hamiltonian Operators

The following topics in the theory of Hamiltonian operators are considered in this thesis.1. Based on the new concepts, formal adjoint and formal closure of a unbounded block operator matrix, general approach to the multiplication or adjoint operation of2x2block operator matrices with unbounded entries are founded, criteria for self-adjointness of block operator matrices based on their entry operators are established and, furthermore, classifications of self-adjointness of block operator matrices are obtained.2. Symplectic self-adjointness of Hamiltonian operator matrices is studied, which is important to 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 as-sumptions, sufficient conditions based on perturbation method are obtained. The theory is applied to a problem in symplectic elasticity.3. Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamil-tonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.4. The concept of symplectic self-adjoint Hamiltonian operator matrices is extended to a wider class of linear operators and, furthermore, for which the properties of numerical ranges, one-parameter unitary semigroups, and maximal Tseng generalized inverses are established.5. Two basic problems in theory of symplectic elasticity, i.e., symplectic orthogonal basis and basis property of the system of generalized eigenfunctions of a Hamiltonian operator, are studied. The structure of a symplectic orthogonal basis is established. A new concept, pre-complete Hamiltonian operator, is suggested; the corresponding expansion theorems are proved. The theory is applied to problems in mathematical physics as well as in elasticity.