The Spectrum And Numerical Range Of Some Special Operator Matrices

In recent years,the spectral theory of linear operator is one of the active research topics. Because the spectral theory of linear operator can solve the problems, which includes the mathematics, physics and its application field, such as bending equation and oscillation equation.Firstly, non-self-adjoint operators play an important role in practical application. But it is very difficulty to study the properties of non-self-operators. So many scholars have studied some special non-self-adjoint operators, including the infinite dimensional Hamiltonian operator、the infinite dimensional skew-Hamiltonian operator and the J-self-adjoint operator and so on. The α-J self-adjoint operator are more extensive than the infinite dimensional Hamiltonian operator and the J-self-adjoint operator. According to the research progress of the infinite dimensional Hamiltonian operator, the symmetry of Fredholm spectrum of α-J self-adjoint operator is obtained. According to the particulary of α-J self-adjoint operator, the Fredholm spectrum of α-J self-adjoint operator is further refined.Secondly, a class of non-self-adjoint operators is obtained on the basis of the in-finite dimensional Hamiltonian operator and α-J self-adjoint operator. Point spec-trum、 residual spectrum and continuous spectrum of the kind of operator are studied. The symmetries with respect to a line for its first-type point and first-type residual spectrum, second-type point spectrum and second-type residual spectrum, third-type point spectrum,and fourth-type point spectrum are given. And the examples are pre-sented to verify the correctness of the conclusion.Finally, on the basis of the last chapter, the numerical range and quadratic nu-merical range of this kind of operators are researched. The symmetry of the numerical range and quadratic numerical range, and the relation that closure of the numerical range and quadratic numerical range contains the spectrum are establised.