Title: 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 $1\leq p<n$, by modifying Voiculescu's invariant we
give examples of commuting $n$-tuples of self-adjoint operators in
$\mathcal{M}$ that are not arbitrarily small perturbations of commuting
diagonal operators modulo $\mathcal{M}\cap L^{p}\left(\mathcal{M},\tau\right)$.