On inverses of Krein's Q-functions

Claudio Cacciapuoti; Davide Fermi; Andrea Posilicano

paper

On inverses of Krein's Q-functions

arXiv:1809.05150

Abstract

Let $A_{Q}$ be the self-adjoint operator defined by the $Q$ -function $Q : z \mapsto Q_{z}$ through the Krein-like resolvent formula $(- A_{Q} + z)^{- 1} = (- A_{0} + z)^{- 1} + G_{z} W Q_{z}^{- 1} V G_{z ˉ}^{*}, z \in Z_{Q},$ where $V$ and $W$ are bounded operators and $Z_{Q} := {z \in ρ (A_{0}) : Q_{z} and Q_{z ˉ} have a bounded inverse} .$ We show that $Z_{Q} \neq = \emptyset ⟹ Z_{Q} = ρ (A_{0}) \cap ρ (A_{Q}) .$ 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.