On inverses of Krein's Q-functions

TL;DR

This paper investigates the properties of the inverse of Krein's Q-functions within the context of self-adjoint operators, extending previous results by removing the assumption of a Nevanlinna function representation.

Contribution

It proves that the set where the Q-function is invertible coincides with the intersection of the resolvent sets of the involved operators, without assuming a Nevanlinna function structure.

Findings

Z_Q is non-empty implies Z_Q equals the intersection of the resolvent sets of A_0 and A_Q.

The result extends previous theorems by not requiring Q to be a Nevanlinna function.

The proof uses algebraic manipulations based on the first resolvent identity.

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 $$(-A_{Q}+z)^{-1}= (-A_{0}+z)^{-1}+G_{z}WQ_{z}^{-1}VG_{\bar z}^{*}\,,\quad z\in Z_{Q}\,,$$ 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.