# Perturbations of self-adjoint operators in semifinite von Neumann algebras: Kato–Rosenblum theorem

### Abstract

• In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $\tau$ be a faithful normal semifinite tracial weight of $\mathcal M$. Suppose that $H$ and $H_1$ are self-adjoint operators affiliated with $\mathcal{M}$. We show that if $H-H_1$ is in $\mathcal{M}\cap L^{1}\left(\mathcal{M},\tau\right)$, then the ${norm}$ absolutely continuous parts of $H$ and $H_1$ are unitarily equivalent. This implies that the real part of a non-normal hyponormal operator in $\mathcal M$ is not a perturbation by $\mathcal{M}\cap L^{1}\left(\mathcal{M},\tau\right)$ of a diagonal operator. Meanwhile, for $n\ge 2$ and \$1\leq p
• ### Authors

• Li, Qihui
• Shen, Junhao
• Shi, Rui
• Wang, Liguang

• July 2018

### Keywords

• Primary: 47C15, Secondary: 47L60, 47L20
• math.OA

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