The Right Invertibility And Spectrum Of Anti-Triangular Operator Matrices

For a class of anti-triangular operator matrices M=(?), the denseness and the closedness of their range and the right invertibilitv are studied, and some sufficient and necessary conditions for them to bo characterized by off-diagonal entries are given. Be-sides, the spectrum of anti-triangular operator matrices M= (?) is studied. The properties, including injectiveness. surjectiveness and density of range, are presented by the quadratic complements, and the point spectrum, continuous spectrum and residual spectrum are then described.