Weak type estimates for the absolute value mapping

2013/09/13 by M. Caspers, D. Potapov, Caspers, M. +5 · 2 citations
Mathematics · #47A30 #47B10 #47L20 #Advanced Operator Algebra Research #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1309.3378

openalex publication_date 2013/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if A and B are bounded self-adjoint operators such that A-B belongs to the trace class, then |A| -|B| belongs to the principal ideal L_1,∞ in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert space generated by an operator whose sequence of eigenvalues is 1, 1/2, 1/3, 1/4, .... Moreover, μ(j;|A| -|B|)≤ const(1 + j)^-1‖A-B‖₁. We also obtain a semifinite version of this result, as well as the corresponding commutator estimates.

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