Eigenvalues of singular measures and Connes noncommutative integration

2021/03/02 by Rozenblum, Grigori · 2 citations
#35P20 #47G30 #58J50 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2103.02067

Abstract

For a singular measure μ, Ahlfors regular of order α>0, with compact support in ℝ^N and a pseudodifferential operator A of order -l=-N/2 we consider the compact operator T(P,A) = A^*PA. Here P is the signed measure, P=Vμ with density V belonging to the Orlicz class L^Ψ,μ with Ψ(t)=(t+1)log(t+1)-t. Using eigenvalue estimates for such operators, obtained in arXiv:2011.14877, we establish eigenvalue asymptotics of T(P,A) for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators T(P,A) and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.

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• Singular value asymptotics on compact smooth Riemaniann manifolds
• Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures

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