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

Academic Article


  • 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
  • Status

    Publication Date

  • July 2018
  • Has Subject Area

    Published In


  • Primary: 47C15, Secondary: 47L60, 47L20
  • math.OA
  • Digital Object Identifier (doi)

    Start Page

  • 259
  • End Page

  • 287
  • Volume

  • 275
  • Issue

  • 2