The Quadratic Numerical Range And Spectrum Of Block Operator Matrices

Block operator matrix is a special matrix with linear operator as its entries.It is widely used in the fields of system theory,nonlinear analysis and evolution equation problems.Therefore,the spectral inclusion properties of some block operator matrices have been paid more attention by many scholars.In this thesis,we use the numerical range and the quadratic numerical range to estimate the spectrum of some block operator matrices.Firstly,we study a class of off-diagonally dominant block operator matrices A_±D=（?）,where B is a dense closed operator and D is accretive.For bounded block operator matrices A_±D,we are based on the structure of operator matrices and the properties of its entries,analyze the spectral inclusion properties of A_±Dby using the numerical range and quadratic numerical range respectively.In particular,we give a spectral gap of operator matrices A_±D,and obtain a new spectral enclosures that are much more powerful than classical numerical range bounds.For unbounded block operator matrices A_±D,we show that the approximate point spectrum is contained in the closure of the quadratic numerical range,and achieve some new spectral inclusion properties under certain conditions.Secondly,we study the operator M=（?） which is associated with the second order differential equation （?）（t）+ B（?）（t）-Az（t）= 0 in a Hilbert space,where A is a self-adjoint and uniformly positive linear operator,B is accretive.We prove that M is a boundedly invertible closed operator and （?）=M where H₁= D（A） with the norm ||x||H₁= ||Ax||.And we characterize the spectral distribution of the operator M by using the quadratic numerical range of the block operator matrix M|_H1×H1.