# On inverses of Krein's Q-functions

September 13, 2018

Let $A_{Q}$ be the self-adjoint operator defined by the $Q$-function $Q:z\mapsto Q_{z}$ through the Krein-like resolvent formula $$(-A_{Q}+z)^{-1}= (-A_{0}+z)^{-1}+G_{z}WQ_{z}^{-1}VG_{\bar z}^{*}\,,\quad z\in Z_{Q}\,,$$ where $V$ and $W$ are bounded operators and $$Z_{Q}:=\{z\in\rho(A_{0}):\text{$Q_{z}$ and $Q_{\bar z }$ have a bounded inverse}\}\,.$$ We show that $$Z_{Q}\not=\emptyset\quad\Longrightarrow\quad Z_{Q}=\rho(A_{0})\cap \rho(A_{Q})\,.$$ 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.

open access link

Rendiconti di Matematica e delle sue Applicazioni, 39 (2018), 229-240

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