On the left (right) invertibility of operator matrices.

Let H be a complex separable infinite-dimensional Hilbert space. Given the operators A ∈ B(H) and B ∈ B(H), we define MX := ... 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 ..., where σ* is the left (right) spectrum. Finally, we further characterize the perturbation of the left (right) spectrum for Hamiltonian operators. [ABSTRACT FROM AUTHOR]