Almost commuting functions of almost commuting self-adjoint operators

2014/12/11 by Aleksei Aleksandrov, Aleksandrov, Aleksei, Vladimir Peller +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #math.CA #math.CV #math.FA #math.SP

paper · pdf · doi:10.48550/arxiv.1412.3535

6 pages

arxiv created 2014/12/11 · arxiv updated 2014/12/12

Abstract

Let A and B be almost commuting (i.e, AB-BA∈\bS₁) self-adjoint operators. We construct a functional calculus \f↦\f(A,B) for \f in the Besov class B_\be,1¹(\R²). This functional calculus is linear, the operators \f(A,B) and ψ(A,B) almost commute for \f, ψ∈ B_\be,1¹(\R²), \f(A,B)=u(A)v(B) whenever \f(s,t)=u(s)v(t), and the Helton--Howe trace formula holds. The main tool is triple operator integrals.

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