Research On Spectral Problems Of Tridiagonal Operator Matrices

Tridiagonal operator matrix is a special kind of operator matrix,which has important applications in mathematical physics fields such as partial differential equations and elastic mechanics.The estimation of its spectrum also provides an effective method for solving many problems in the fields of physics and mechanics.Therefore,the spectral theory of the tridiagonal operator matrix has been widely concerned by many researchers.In this thesis,we not only analyze the basic properties of tridiagonal operator matrices,but also study their spectral estimations.First,the basic properties of n × n unbounded tridiagonal operator matrices are analyzed.Firstly,the relative boundedness of operator matrices is studied by analyzing the relationship between the elements of operator matrices.Then,by combing with the properties of internal elements,the sufficient conditions for the operator matrix to be closable(closed)are given.Finally,we study their invertibility.Then,the spectral inclusion properties of unbounded tridiagonal operator matrices are studied.We first characterize the spectral inclusion properties of n × n unbounded tridiagonal operator matrices by using Schur complement.Moreover,we use Schur complement and numerical range to estimate spectrum of some 3×3 unbounded tridiagonal diagonally dominant operator matrices,and obtain some more finer estimations of spectrum.