The Local Spectrum Of Hamilton Operator

The class of operators and the spectral theory of operator are one of the hottest topics in operator theory recently, which are widely used in many fields such as mathematical physics and engineering, and the research of them is related to many branches of pure and applied mathematics. This dissertation mainly studies the local spectral theory of Hamilton operator. In order to study the lo-cal spectrum of Hamilton operator better, first we study the spectral properties of quasi-*-A(n) operator and quasi-*-paranormal operator, the perturbations of property (aR) and property (gaR) and the invariant subspace of algebraic exten-sion of nth root of M-hyponormal operator and algebraic extension of nth root of class A operator, then the local spectral properties of Hamilton type operator and extended Hamilton operator are studies. The main results of this dissertation are as follows:Firstly, the inclusion relations between quasi-*-A(n) operator, quasi-*-parano rmal operator and other classes of operators are studied. The spectral continu-ity, Riesz projection operator and the tensor product of quasi-*-.A(n) operator are considered. The Weyl type theorems of algebraically quasi-*-A(n) operator and algebraically quasi-*-paranormal operator are obtained. And some special properties of quasi-*-paranormal operator which satisfies symplectic self-adjoint are given.Secondly, property (aR) and property (gaR) are introduced, which are vari-ations of Weyl theorem. The relations between property (aR), property (gaR) and other Weyl type theorems, and the perturbations of property (aR), property (gaR) are given.Then, the invariant subspace problem of algebraic extension of nth root of class A operator and algebraic extension of nth root of M-hyponormal operator is considered.Lastly, Hamilton type operator and extended Hamilton operator are intro-duced, which are related to Hamilton operator. The decomposability of them and the similar properties with their adjoint operators are studied. The Weyl type theorems and invariant subspace problem of extended Hamilton operator are studied. As corollaries, some related properties of Hamilton operators are obtained.