On the left (right) invertibility of operator matrices

Let H be a complex separable infinite-dimensional Hilbert space. Given the operators A is an element of B(H) and B is an element of B(H), we define MX := [A X 0 B] where X. S(H) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for MX to be a left (right) invertible operator for some X. S(H). Moreover, it is shown that boolean AND(X is an element of S(H)) sigma*(M-X) = boolean AND(X is an element of B(H)) sigma*(M-X) boolean OR Delta, where sigma* is the left (right) spectrum. Finally, we further characterize the perturbation of the left (right) spectrum for Hamiltonian operators.