Spectra And Completion Problems Of Unbounded Operator Matrices

This work mainly deals with the spectral properties and completion problems of un-bounded operator matrices in Hilbert spaces. More precisely, some spectra of unbounded upper triangular operator matrices are described by the corresponding spectra of diagonal operators. Then, asymptotic estimation of the point spectra of some Hamiltonian opera-tors is given and completion problems of unbounded upper triangular operator matrices are studied. The main results of this dissertation are as follows:First, in order to study unbounded upper triangular operator matrices, we discuss the bounded case. Namely, the bounded operator matrix MC=(0ABC) is studied. The necessary and sufficient condition under which the essential spectrum (resp., the Weyl spectrum, the Browder spectrum, the essential approximate point spectrum, the Browder essential approximate point spectrum) of MC is equal to the union of the essential spec-trum (resp., the Weyl spectrum, the Browder spectrum, the essential approximate point spectrum, the Browder essential approximate point spectrum) of diagonal entries A, B is given.Next, the unbounded operator matrix TB=(0ABD) is considered and the necessary and sufficient condition under which the essential spectrum (resp., the Weyl spectrum, the Browder spectrum, the approximate point spectrum, the defect spectrum) of TB coincides with the union of the essential spectrum (resp., the Weyl spectrum, the Browder spectrum, the approximate point spectrum, the defect spectrum) of diagonal entries A, D is given. As applications, the spectra properties of upper triangular Hamiltonian operator matrices are obtained.Then, the properties of the point spectrum of some classes of Hamiltonian operator matrices are studied. The upper and lower bounds for the point spectrum of off-diagonal Hamiltonian operator matrices are given and the upper or lower bound for the point spectrum of diagonally dominant Hamiltonian operator matrices is obtained by using the min-max principle. Moreover, the theory is applied to several concrete examples arising in mathematical physics.Finally, the completion problems of the unbounded upper triangular partial operator matrix (0AD?) are discussed. For given closed operators A, D with dense domains, the necessary and sufficient conditions are obtained, respectively, for the unbounded upper triangular operator matrix TB to be a semi-Weyl operator and a semi-Fredholm operator for some closable operator B. Then the intersection of residual spectra (resp., continuous spectra, closed range spectra) and the union of closed range spectra of TB are accurately described. If A is a bounded linear operator, we also obtain the intersection of point spectra and the union of residual spectra (resp., continuous spectra) of TB.