The Spectral Analysis Of Some Block Operator Matrices

The block operator matrix is one of the main contents of linear operator theory.It has important applications in the fields of mathematical physics,e.g.functional analysis,coupled systems of partial differential equations,elastic mechanics,fluid me-chanics and etc.The spectral theory of the block operator matrix,such as spectral structure and spectral distribution,describes not only the essential effect of the lin-ear operator,but also the characteristics and stability of the system energy changes Therefore,the spectral theory of block operator matrices has been widely concerned by many researchers.In this paper,we discuss the spectral inclusion properties of the block operator matrices from two different perspectivesFirstly,we discuss the spectral estimation of N × N order bounded block operator matrices.Based on the Brauer-Ostrowski theorem of the approximate point spectrum of bounded block operator matrices in[19],we give the generalized Brauer-Ostrowski theorem of its approximate point spectrum,and obtain its more detailed estimation for approximate point spectrum.Then,on the basis of the above approximate point spectral inclusion properties,we characterize the spectral inclusion properties of the bounded block operator matrices,and obtain the generalized Brauer-Ostrowski the-orem of the spectrum of bounded block operator matrices and the related spectral estimation theorems,and then obtain the Brauer theorem,Ostrowski theorem,gener-alize the Gershgorin's theorem for block operator matricesSecondly,we study the unbounded linear operator A=(?)which is asso-ciated with the second order differential equation x(t)+Dx(t)+A0x(t)=0,where A0 is an uniformly positive self-adjoint operator in the Hilbert space H.Based on the main results in[31],we estimate the spectral range of unbounded operator A us-ing the supremum and infimum of the real part of the numerical range of D and the quadratic numerical range of the block operator matrix A|H1×H1,where A|H1×H1=A?H1=D(A0)is a Hilbert space,and obtain more accurate estimation of spectrum.