A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.
Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian 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.
Given two closed, in general unbounded, operators A and C, we investigate the left invertible completion of the partial operator matrix A ？ 0 C. Based on the space decomposition technique, the alternative sufficient and necessary conditions are given according to whether the dimension of R（A）⊥ is finite or infinite.As a direct consequence, the perturbation of left spectra is further presented.
In this paper, the authors investigate the spectral inclusion properties of the quadratic numerical range for unbounded Hamiltonian operators. Moreover, some examples are presented to illustrate the main results.
