Functions of almost commuting operators and an extension of the Helton-Howe trace formula

2015/08/19 by А. Б. Александров, Alexei Aleksandrov, Aleksandrov, Alexei +2
Mathematics · #47A60 #47B10 #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CA #math.CV #math.FA #math.SP #msc:47A60 #msc:47B10

paper · pdf · doi:10.48550/arxiv.1508.04702

20 pages. arXiv admin note: substantial text overlap with arXiv:1505.07173

arxiv created 2015/08/19 · openalex publication_date 2015/08/19 · arxiv updated 2015/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A and B be almost commuting (i.e., the commutator AB-BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ↦φ(A,B) for functions φ in the Besov class B_∞,1¹(\Bbb R²). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ, ψ∈ B_∞,1¹(\Bbb R²), and φ(A,B)=u(A)v(B) whenever φ(s,t)=u(s)v(t). We extend the Helton--Howe trace formula for arbitrary functions in B_∞,1¹(\Bbb R²). The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L^∞ spaces.

Citations

• Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals

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