Research On The Generalized Drazin Spectrum Of The Upper Triangular Operator Matrix

Research on various spectral theories of the 2×2 upper triangular operator matrix has attracted the attention of many scholars.Since the operator matrix has a certain connection with its internal operators,the spectral of the upper triangular operator matrix can be characterized by the spectrum of diagonal operators.The spectral hole filling and spectral perturbation problems of the upper triangular operator matrix is studied in this thesis.This thesis has been divided four parts.In the first part,preliminary knowledge is introduced.The related concepts and basic theories of Banach space and Hilbert space of bounded linear operators,Banach algebra are introduced.In the second part,the operator matrices(?)on Banach algebra are studied.First,the relationships of the generalized Drazin spectrum among operators Mc and a,b are obtained.That is ?gd(a)??gD(b)=?gD(Mc)?Wg and(?).Second,other properties of the generalized Drazin spectrum are also studied.In the third part,the perturbation problems of operator matrices(?)on Banach space are mainly studied.First,given operators A and B a necessary and sufficient condition for Mc to be generalized Drazin invertible for some operators C?B(Y,X)are given.Second,the set(?)is completely characterized.Third,the relationship between pseudo B-Browder spectrum and generalized Drazin spectrum is studied.In the fourth part,the operator matrices(?)on Banach space are studied.First,given operators A and B,a necessary and sufficient condition for Mc to be B-Browder operator for some C?B(Y,X)is given.Second,the set(?)is completely characterized.