On properties of the \(\xi \)-function in semi-finite von Neumann algebras
Let \(\mathcal A\) be a semi-finite von Neumann algebra and \(\tau\) a normal faithful semifinite trace on it. One of the main objectives of the present paper is to prove versions of the Birman-Schwinger principle for (relative) trace class perturbation problems of dissipative operators in \(\mathcal A\) and selfadjoint operators affiliated with \(\mathcal A\). E.g., it is shown that for bounded and boundedly invertible \(M=M^*\) in \(\mathcal A\), a selfadjoint operator \(H_0\) affiliated with \(\mathcal A\) and \(K\) in the \(\tau\)-trace ideal \(\mathcal L_2(\mathcal A,\tau)\), the relation \[ \xi\bigl(\lambda,H_0,H_0+KM^{-1}K^*\bigr)=\lim_{\varepsilon\rightarrow 0+} \xi\bigl(M+K^*(H_0-\lambda-i\varepsilon)^{-1}K,M\bigr) \] holds for a.e.\ \(\lambda\in\mathbb R\). This principle is then further generalized to the case when \(\lambda\) is in the spectrum of the perturbed or unperturbed operator. Moreover, if the normal boundary values \(K^*(H_0-\lambda-i0)^{-1}K\) exist in the operator norm with imaginary parts in the \(\tau\)-trace class, the function \(\xi(\lambda,H_0,H_1)\) is related to the scattering operator associated with the pair \(\{H_0-\lambda,H-\lambda\}\).
scientific article; zbMATH DE number 5617569