Completion Problems And Spectra For Operator Matrices

Operator matrices are heated topic in operator theory, and the research of operator matrices has related to pure and applied mathematics such as matrix analysis, optimality principle and quantum physics etc. This dissertation mainly studies completion problems and spectra of operator matrices, including spec-tral completion problems, Fredholm completion problems, invertible completion problems of operator matrices and the structure of spectra of nonselfadjoint op-erator matrices.We use H1 and H2 to denote Hilbert spaces. Let B(Hi,Hj) be the Banach space of all bounded linear operators from Hi to Hj, and write B(Hi, Hi)= B(Hi), i,j= 1,2. The main results of this dissertation are as follows:Firstly, completion problems of the Moore-Penrose spectrum for an operator partial matrix (?) are discussed. For given operators A∈B(H1) and B∈B(H2), the following sets are characterized, whereσm(·) denotes the Moore-Penrose spectrum.Secondly, right (left) Fredholm completion problems and right (left) invert-ible completion problems for an operator partial matrix (?) are studied. For given operators A∈B(H1), B∈B(H2) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator and a right (left) invertible operator for some X∈B(H1,H2). Then, Fredholm completion problems and invertible completion problems for an operator partial matrix (?) are considered. For given operators A∈B(H1) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator, a right (left) invertible operator, a Fred-holm operator and an invertible operator for some X∈B(H1,H2) and Y∈B(H2).Lastly, the structure of spectra for some class of nonselfadjoint operator matrices is studied. As corollaries, some related properties of infinite dimen-sional Hamiltonian operators and J-selfadjoint operator matrices are obtained. In general, infinite dimensional Hamiltonian operators are a class of special non-selfadjoint operator matrices, its study has important value in both theory and applications. Therefore, the approximate point spectrum of off-diagonal infinite dimensional Hamiltonian operators and the essential spectrum of upper trian-gular infinite dimensional Hamiltonian operators are considered. A relationship between the approximate point spectrum (or the essential spectrum) of infinite dimensional Hamiltonian operators and the approximate point spectrum (or the essential spectrum) of their entries is given.