On inverses of Kreın’s Q-functions

Let AQ be the self-adjoint operator defined by the Q-function Q: z 7→ Qz through the Kreın-like resolvent formula (−AQ + z)−1 = (−A0 + z)−1 + GzWQ−z1V G∗z¯ , z ∈ ZQ , where V and W are bounded operators and ZQ := {z ∈ ρ(A0): Qz and Qz¯ have a bounded inverse}. We show that ZQ 6= ∅ = ZQ = ρ(A0) ∩ ρ(AQ) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.