The Spectrum Of Upper Triangular Infinite Dimensional Hamiltonian Operators And Applications

Infinite Hamiltonian operators are originated from infinite-dimensional Hamilto-nian system, which is engaged in profound mechanical background. This dissertation mainly investigates the spectrum and invertibility of the upper triangle infinite dimensional Hamiltonian operators and applications.All along, spectral theory of non-selfadjiont operators did not develop a satisfactory theoretical framework. Non-selfadjiont operators which is engaged in profound mechanical background: infinite dimensional Hamiltonian operators, J-selfadjiont operators, transport operators, have not uniform way about their spectral investigation. In this dissertation , firstly the definitions of operator of same point spectrum and operator of opposite point spectrum are given. J-selfadjiont operators and transport operators are all operators of same point spectrum, infinite dimensional Hamiltonian operators are all operators of opposite point spectrum. The description of residual spectrum of the closed densely defined operators in Hilbert space is obtained, by the above description, a necessary and sufficient condition of the residual spectrum of the closed densely defined operators in Hilbert space being empty is given; by the result, the spectral structures of operator of same point spectrum and operator of opposite point spectrum are given; by the results, the spectral structures of the above non-selfadjiont operators are obtained.In order to pratical applications, the spectral description of upper triangle infinite dimensional Hamiltonian operators is studied. Infinite dimensional Hamiltonian operators are a class of special 2×2 operator matrices, by its spectral structure and structure character, we get the necessary and sufficient condition of the spectrum of upper triangle infinite dimensional Hamiltonian operators with diagonal domain being equivalently described by the spectrum of the first diagonal element. by the result, the spectral theoretical mode of plane elasticity problems without the body force is established. And we obtain the invertibility and description of spectrum and continuous spectrum of upper triangle infinite dimensional Hamiltonian operators with upper dominant.In addition, we obtain the sufficient conditions of algebric index of the pure imaginary piont spectrum of a class of infinite dimensional Hamiltonian operators being one.Nonnegative Hamiltonian operators have important application in linear quadratic optimal control problem. The inverbility and distribution of point spectrum of a class of nonnegative Hamiltonian operators are obtained.