Spectrum Of Off-Diagonal Operator Matrices

To begin with, the research of the spectrum of Off-Diagonal operator matrices is briefly outlined. Besides, Relation between approximate point spectrum, detect spec-trum,essential spectrum of T= which is off-diagonal closed linear operator matrices acting in the Hilbert space X (?) X and the product of internal elements operator BC,CB is studied. By using the properties of ap-proximate point spectrum and detect spectrum given the relation between the spectrum T and spectrum of the product of BC.CB. As an application,calculate the spectrum of the wave equation and the harmonic equation.and the essential spectrum of Hamiltonian operator matrix which is derived from the plate bending equation is discussed.