1. Hamiltonian Systems.- 1.1 Finite-dimensional Hamiltonian Systems and Hamiltonian Matrices.- 1.2 Linear Infinite-dimensional Hamiltonian Canonical Systems and Infinite-dimensional Hamiltonian Operators.- 1.3 Hamiltonian Canonical Systems and the Pseudo-division Algorithm of Multivariate Polynomial Matrices.

- 1.4 Monic Factorization of Bivariate Polynomial Matrices.- 1.5 Matrix Method of Solving Characteristic Sequences of the System of Polynomials.- 2 Spectra of Infinite-dimensional Hamiltonian Operators.- 2.1 Spectra of Linear Operators.- 2.

2 Spectra of Diagonal Infinite-dimensional Hamiltonian Operators.- 2.3 Spectra of Skew-diagonal Infinite-dimensional Hamiltonian Operators.- 2.4 Spectra of Upper Triangular Infinite-dimensional Hamiltonian Operators.- 2.5 Spectra of Non-negative Hamiltonian Operators.- 2.

6 Spectra of the General Infinite-dimensional Hamiltonian Operators.- 3 Completeness of Eigenvector System of Infinite-dimensional Hamiltonian Operators.- 3.1 Infinite-dimensional Symplectic Spaces.- 3.2 Symplectic Orthogonality of Eigenvector System of Infinite-dimensional Hamiltonian Operators.- 3.3 Completeness of Eigenfunction System of 2×2 Infinite-dimensional Hamiltonian Operators.

- 3.4 Completeness of the Eigenfunction System of 4×4 Infinite-dimensional Hamiltonian Operators.- 4 Symplectic Self-adjointness of Infinite-dimensional Hamiltonian Operators.- 4.1 Definition of Symplectic Self-adjoint Operators.- 4.2 The Adjoint of the Sum of Two Operators.- 4.

3 The Adjoint of Product of Operators.- 4.4 Characterization of Symplectic Self-adjointness by Using the Spectral Set of Infinite-dimensional Hamiltonian Operators.- 5 Numerical Range Theory of Infinite-dimensional Hamiltonian Operators.- 5.1 Numerical Range and Its Definition.- 5.2 Numerical Range of Infinite-dimensional Hamiltonian Operators.

- 5.3 Numerical Radius of Infinite-dimensional Hamiltonian Operators.- 5.4 Quadratic Numerical Range of Infinite-dimensional Hamiltonian Operators.- 5.5 Quadratic Numerical Radius of Infinite-dimensional Hamiltonian Operator.- 5.6 Essential Numerical Range of Infinite-dimensional Hamiltonian Operators.

- 6 Spectral Theory of Infinite-dimensional Hamiltonian Operators in Complete Indefinite Inner Product Spaces.- 6.1 Krein Spaces.- 6.2 Spectra of Infinite-dimensional Hamiltonian Operators in Krein Spaces.- 6.3 ℑ−numerical Range in Krein Spaces.