Weyl Spectrum Of Off-diagonal Operator Matrices

For the off-diagonal operator matrix H =(OBCO):D(C)(?)D(B)(?)X(?)→X(D X,the(left,right)Weyl spectrum of H is given,based on the spectral properties of the product BC and CB of its diagonal entries.Moreover,we study the symmetry of the Weyl spectrum of off-diagonal Hamiltonian operators,and farther obtain a necessary and sufficient condition for the Weyl spectrum of general Hamiltonian operator to be symmetric with respect to the real axis.