The Spectrum And Quadratic Numerical Range Of Operator Matrices

It is important for spectral theory of linear operator to apply to mechanics, magnetohydrodynamics, quantum mechanics, biology and engineering technology, etc.Firstly, the applied value of the spectral theory of infinite-dimensional Hamilton operator as a special linear operator which is in the practical problems such as mechanics, physics and so on, as well as the important role of the approximate point spectrum for the distribution of spectral estimation is considered. Based on the thought of operator matrix and the subdivision of point spectrum and residual spectrum, a sufficient condition, which first-type point spectrum, first-type residual spectrum and approximate point spectrum for upper triangular infinite dimensional Hamiltonian operator with diagonal domain is equal to the corresponding spectrum of its first diagonal element respectively, is obtained. The properties of approximate point spectrum for upper triangular infinite dimensional Hamiltonian operator which its first diagonal element is the special operator is given. Combining with the specific applications in mechanics,the reasonableness of the conclusions is shown.Secondly, the non-self-adjoint operators did not develop a perfect theoretical framework, some special operators which are engaged in profound mechanical background was merely studied. Based on the idea of unity, the symmetry of a line through origin for first-type point spectrum with first-type residual spectrum, second-type point spectrum with second-type residual spectrum, third-type point spectrum, and fourth-type point spectrum of unbounded - self-adjoint operator as a class of non-self-adjoint operators is obtained, respectively. While the corresponding spectral properties of infinite dimensional Hamiltonian operator, infinite dimensional Skew-Hamiltonian operator and self-adjoint operator which are the special unbounded - self-adjoint operators are also described and effectively validated with the instance.Finally, It is important significant for the numerical range and quadratic numerical range to characterize the spectral range, based on the idea of comparison, the properties of the numerical range and the quadratic numerical range of a class of infinite-dimensional Skew-Hamilton operators are studied. These infinite-dimensional Skew-Hamilton operators which are in the Skew-Hamilton system are widely used in some problems such as mechanics, mathematical physics, optimization, and so on.The symmetry of the numerical range and quadratic numerical range with respect to the real axis respectively is given. The relationship between Spectrum and numerical range or quadratic numerical range is obtained, and the rationality of the conclusion is proved by using the example of elasticity mechanics. Furthermore, the similarities and differences of the properties for numerical range and quadratic numerical range are given.