Local Spectral Properties Of Hamilton Operators

In this thesis,we mainly study the local spectral properties of Hamilton operators.In order to study the local spectral properties of Hamilton operators,we first introduced and studied two Weyl type theorems-property(h)and property(gh),and consider their perturbation problems and the class problems of bounded operator satisfying these Weyl type theorems.Some local spectra(such as the upper semi-Browder spectrum and the upper semi-Weyl spectrum)of unbounded upper triangular operator matrices are studied by using spatial decomposition method.The local spectra of unbounded upper triangular operator matrices are characterized by the local spectra of diagonal operators.For bound-ed upper triangular operator matrices,the upper semi-B-Fredholm spectrum?the lower semi-B-Fredholm spectrum and the B-Fredholm spectrum are studied,sufficient conditions are obtained for upper semi-B-Fredholm spectrum(lower semi-B-Fredholm spectrum and B-Fredholm spectrum)of bounded upper triangular operator matrices equal to union of upper semi-B-Fredholm spectrum(lower semi-B-Fredholm spectrum and B-Fredholm spectrum)of diagonal operator respectively.Finally,some local spectral properties(such as strongly decomposable)of Hamilton operators are studied.The specific contents are as follows:First,we introduced two Weyl type theorems-property(h)and property(gh),established for a linear bounded operator defined on Banach space several sufficient and necessary conditions for which property(h)and property(gh)hold,studied their relations with other Weyl type theorems(such as Weyl theorem and a-Weyl theorem)and their perturbations,and find out the bounded operator classes satisfying property(h)and property(gh).The perturbation problem of property(h)and property(gh)is solved.Secondly,the spectral properties of unbounded triangular operator matrix are stud-ied.The necessary and sufficient conditions for the upper semi-Browder spectrum and the upper semi-Weyl spectrum of unbounded upper triangular operator matrix to be equal to the union of the corresponding spectra of the diagonal operator are obtained.As an application,the spectral properties of Hamilton operator matrix are obtained.Then,we studied the upper semi-B-Fredholm spectrum?the lower semi-B-Fredholm spectrum and the B-Fredholm spectrum?the Weyl type theorems of bounded triangular operator matrices and obtained sufficient conditions for the equivalence of several kinds of Weyl type theorems(such as Weyl theorem and a-Weyl theorem)of bounded triangular operator matrices.These conditions are described by the properties of sub-block opera-tors.Sufficient conditions are obtained for the upper semi-B-Fredholm spectrum(the lower semi-B-Fredholm spectrum and the B-Fredholm spectrum)of a bounded upper triangu-lar operator matrix to be equal to the union of the upper semi-B-Fredholm spectrum(the lower semi-B-Fredholm spectrum and the B-Fredholm spectrum).As an application,the corresponding properties of Hamilton operator matrix are obtained.Finally,some local spectral properties of Hamilton type operators and extended Hamilton operators are studied.The strongly decomposable,Weyl type theorems and hyper invariant subspaces of Hamilton type operators and extended Hamilton operators are considered.The strongly decomposable,Weyl type theorems,hyper invariant subspaces and similar properties of conjugate operators of Hamilton type operators and extended Hamilton operators are obtained.As an application,the corresponding local properties of Hamilton operator are obtained.