Completion Problems Of Operator Matrices And Spectral Of Infinite Dimensional Hamilton Operators

There have some connections between the point spectrum and the residual spectrum of linear operator, by dividing the point spectrum and the residual spectrum into more classes,more and better properties of spectral of linear operator can be obtained. So the point spectrum was divided into two classes, in this paper, denote byσpa(·) andσpb(·). But, in the couse of studying the spectral completion problems of operator matrices, we find that the residual spectrum and the point spectrum are necessary to be further classi-fied, we divide the residual spectrum into two kinds denote byσr1(·) andσr2(·), divideσpa(·) into two kinds denote byσp1(·) andσp2(·) and divideσpb(·) into two kinds denote byσp3(·) andσp4(·), now the point spectrum is divided into four kinds. This disserta-tion mainly investigates completion problems of bounded operator matrices, including spectral completion problems of four kinds of point spectrums and two kinds of residual spectrums and so on, the spectra and spectral completion problems of infinite dimensional Hamiltonian operators.The main results of this thesis are followings:Firstly, spectral completion problems of 2 x 2 upper triangular operator matrices are studided. Let H. and K be Hilbert spaces, and B(H,K) denote the set of bounded linear operators from H to K, with B(H,H) abbreviated to B(H). Let A∈B(H), B∈B(K) are given, define MC=(?). The following sets where t∈{pa,pb,p1,p2,p3,p4,r1,r2}, are characterized, respectively. In addition, the possible spectrum of four kinds of point spectrumσp1(·),σp2(·),σp3(·),σp4(·) the left(right) Weyl spectrum and the Weyl spectrum of MC are discussed, respectively.Secondly, spectral completion problems of 3 x 3 upper triangular operator matrices are considered. The perturbation of the defect spectrum,the approximate point spectrum and the Moore-Penrose spectrum of 3 x 3 upper triangular operator matrices are obtained, respectivelyThen, the " Question 3 " that was arised by Professor Du Hongke in 1994 is discussed. The set of operators C∈B(K,H) valided the " Question 3 " is described. Which lays the necessary foundations for further researches on the " Question 3 ".Lastly, spectra and spectral completion problems of infinite dimensional Hamiltonian operators is studied. Infinite dimensional Hamiltonian operators is a class of special nonselfadjoint operator matrices, its studying has important value in both theory and applications. Theories of spectral of infinite dimensional Hamiltonian operators play an important role in studying of infinite dimensional Hamiltonian systems. So in this paper, properties of spectral of infinite dimensional Hamiltonian operators are considered, including spectral of four kinds of point spectrums and two kinds of residual spectrums and so on. In the general case, infinite dimensional Hamilton operators are unbounded operator matrices, there exist spectral completion problems, but studying of the spectral completion problems of infinite dimensional Hamilton operators is difficulty.By taking advantage of the results of spectral completion problems of bounded operator matrices, we limit the scope of the spectral completion of upper triangle infinite dimensional Hamilton operators with diagonal domain,and obtain the perturbation ofσp1(·) andσr1 (·) of upper triangle infinite dimensional Hamilton operators with diagonal domain. Furthermore, the spectral theoretical of two classes of special infinite dimensional Hamilton operators are considered, respectively.