Closed Range Property Of Symplectic Symmetric Hamiltonian Operators

Using the perturbation theory and the factorization technique of operator matrices,we investigate the closedness and closedness of range for symplectic symmetric Hamiltoni-an operators.The closedness is completely described for the upper-triangular symplectic symmetric Hamiltonian operator with diagonal domain,and,for the general domain case,the characterization of closedness is given under certain assumptions.Next,the condi-tions for the general symplectic symmetric Hamiltonian operator to be closed are studied.Finally,Some descriptions on closedness of ranges are given for the cases such as diagonal dominant and upper dominant,and the results for the general case are actually presented.