Spectral Inclusion And Semigroup Generation Properties Of Unbounded Operator Matrices

This dissertation deals with the spectral inclusion and semigroup generation proper-ties of unbounded operator matrices in Hilbert spaces. The spectral inclusion by numerical range and quadratic numerical range is discussed on the core of its entries. The semigroup generation is studied by using the space decomposition method, quadratic complements and row operators.First, the conditions for the numerical range of the Hamiltonian operator matrix H=(C-AAB) to be symmetric with respect to the imaginary axis are discussed. Based on the symmetry of the numerical range, a necessary and sufficient condition for H to generate C0 semigroups is given. Besides, it is shown that the quadratic numerical range of the Hamiltonian operator matrix is symmetric with respect to the imaginary axis on certain natural assumptions. Then, the spectral inclusion by the quadratic numerical range of such operator is further described.Next, the semigroup generation of the Hamiltonian operator matrix H= (C-AAB) is studied. For the symplectic self-adjoint Hamiltonian operator matrix H, the necessary and sufficient conditions for generating Co semigroups are given by means of the symmetry of its spectrum, and the spectral distribution of H is also determined. In the sequel, we obtain the necessary and sufficient conditions that the upper dominant Hamiltonian operator matrix generates a contraction semigroup.Then, the semigroup generation of a special class of operator matrices M= (CDII) is considered. The necessary and sufficient conditions for M with different domains to generate contraction semigroups are given. Using this result, we consider the operator ma-trix derived from rectangular plates with two opposite sides simply supported in elasticity theory. It is proved that this operator matrix generates contraction semigroup on some Hilbert space. By using the semigroup method, the analytical solutions to the problem are then given.Finally, the semigroup generation of the unbounded operator matrix M = (CDAB) is discussed. As operator matrices with different structures need different space decomposi-tions, we first investigate the unbounded anti-triangular operator matrix M= (C0AB), and obtain the necessary and sufficient conditions for M to generate C0 semigroups. Then, the general unbounded operator matrix M= (CDAB) with natural domain is studied, and the necessary and sufficient conditions for generating C0 semigroups are given.