On the left (right) invertibility of operator matrices

Abstract

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