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

In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let M be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space H and let T be a faithful normal semifinite tracial weight of M. Suppose that H and H-1 are self-adjoint operators affiliated with M. We show that if H Hi is in M boolean AND L-l (M, T), then the norm absolutely continuous parts of H and H-l are unitarily equivalent. This implies that the real part of a non-normal hyponormal operator in M is not a perturbation by M boolean AND L-1 (M, T) of a diagonal operator. (C) 2018 Elsevier Inc. All rights reserved.