Basic Properties Of Block Operator Matrices

The block operator matrix is a matrix with linear operators as its elements.In recent years,the study of the block operator matrix is very active in the field of operator theory.The study of the block operator matrix is of great significance both theoretically and practically.In this dissertation,we mainly consider the conjugate operator and the spectra of the block operator matrix,including the conjugate operator of the block operator matrix,the conjugate operator of the infinite dimensional Hamiltonian operator,the essential spectrum and the Weyl spectrum of the block operator matrix,the essential spectrum and the Weyl spectrum of the infinite dimensional Hamiltonian operator,etc.First,we study the conjugate operator of the bounded 2×2 block operator matrix (?)and discuss the relationship between the conjugate operator of A in the Hilbert space and the conjugate operator of A in the Banach space.Second,the conjugate operator of the unbounded 2×2 block operator matrix A =(CDAB)is discussed.For the unbounded block operator matrix A,the relationships betweena ?(C*D*A*B*)and ?(C'D'A'B')are given.Third,the symplectic self-adjointness of the infinite dimensional Hamiltonian op-erator(i.e.(JH)*= JH)is studied.The necessary and sufficient conditions for the symplectic self-adjointness of the infinite dimensional Hamiltonian operator are obtained:If ???(C),then(JH)*= JH<=>B + ? ? A(C-?)1A*=[B + ? + A(C-?)1A*]*;If ???(B),then(JH)*= JH<=>C + ? + A(B-?)-A*=[C + ? + A(B-?)-1A*]*.Then,for the bounded 2×2 block operator matrix and the unbounded 2×2 block operator matrix,the essential spectrum and the Weyl spectrum are discussed,respectively.The essential spectrum relationships and the Weyl spectrum relationships between the block operator matrix A =(CDAB)and its element operators A,B,C,D are obtained:?e(A)(?)(?e(A)U ?e(D));?w(A)(?)(?w(A)U ?w(D)).Finally,we study the quadratic complement and the Schur complement of the infi-nite dimension Hamiltonian operat ?=(C-*AB),and give the essential spectral relationships and the Weyl spectral relationships between the infinite dimension Hamil-tonian operator H and its element operators A,B,C,-A*,respectively.We obtain the virtual axisymmetric properties of the quadratic complement and the Schur complement of the infinite dimension Hamiltonian operator H:?e(?)\?(-A*)= {??C:S1(?)is not a Freholm operator};?w(?)\?(-A*)= {??C:s1(?)is not a Weyl operator};?e(H)\?(-A*)is virtual axisymmetric;?w(H)\?(-A*)is virtual axisymmetric.