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

2017/06/29 by Qihui Li, Li, Qihui, Junhao Shen +5
Mathematics · #47L20 #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA) #Primary: 47C15 #Random Matrices and Applications #Secondary: 47L60 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1706.09566

openalex publication_date 2017/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 τ be a faithful normal semifinite tracial weight of \mathcal M. Suppose that H and H₁ are self-adjoint operators affiliated with M. We show that if H-H₁ is in M∩ L¹(M,τ), then the norm absolutely continuous parts of H and H₁ are unitarily equivalent. This implies that the real part of a non-normal hyponormal operator in \mathcal M is not a perturbation by M∩ L¹(M,τ) of a diagonal operator. Meanwhile, for n≥ 2 and 1≤ p

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