The intersection of the spectra of operator completions

Let A B(H), B B(K), C B(K,H), X B(H,K) and be an operator completion of the partial operator matrix . In this note, we consider the intersection of the spectra of MX when X runs over B(H,K). Denote by ∑(A,B,C) the set of scalar such that either (A−λ,C) or is not right invertible. We prove that where Δ(A,B,C) is the set of scalars such that R((A−λ,C))=H, , and ind(A−λ)+ind(B−λ)≠0. We also prove that the intersection is empty if and only if (A,C) and (B*,C*) are controllable.