JournalsjstVol. 12, No. 1pp. 259–300

Eigenvalues of singular measures and Connes’ noncommutative integration

  • Grigori Rozenblum

    Chalmers University of Technology, Göteborg, Sweden; The Euler Int. Math. Inst., St. Petersburg, Russia
Eigenvalues of singular measures and Connes’ noncommutative integration cover
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Abstract

In a domain ΩRN\Omega\subset \mathbb{R}^{\operatorname{N}} we consider compact, Birman–Schwinger type operators of the form TP,A=APA\operatorname{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A} with PP being a Borel measure in Ω,\Omega, containing a singular part, and A\mathfrak{A} being an order N/2-\operatorname{N}/2 pseudodifferential operator. Operators are defined by means of quadratic forms. For a class of such operators, we obtain a proper version of H. Weyl's law for eigenvalues, with order not depending on dimensional characteristics of the measure. These results lead to establishing measurability, in the sense of Dixmier–Connes, of such operators and the noncommutative version of integration over Lipschitz surfaces and rectifiable sets.

Cite this article

Grigori Rozenblum, Eigenvalues of singular measures and Connes’ noncommutative integration. J. Spectr. Theory 12 (2022), no. 1, pp. 259–300

DOI 10.4171/JST/401