TY - JOUR

T1 - Eigenvalues of singular measures and Connes’ noncommutative integration

AU - Rozenblum, Grigori

N1 - Grigori Rozenblum, Eigenvalues of singular measures and Connes’ noncommutative integration. J. Spectr. Theory 12 (2022), no. 1, pp. 259–300

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In a domain Ω RN we consider compact, Birman–Schwinger type operators of the form TP;A D A*P A with P being a Borel measure in Ω; containing a singular part, and A being an order -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.

AB - In a domain Ω RN we consider compact, Birman–Schwinger type operators of the form TP;A D A*P A with P being a Borel measure in Ω; containing a singular part, and A being an order -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.

KW - Eigenvalue distribution

KW - noncommutative integation

KW - singular measures

UR - http://www.scopus.com/inward/record.url?scp=85128214705&partnerID=8YFLogxK

U2 - 10.4171/JST/401

DO - 10.4171/JST/401

M3 - Article

AN - SCOPUS:85128214705

VL - 12

SP - 259

EP - 300

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

SN - 1664-039X

IS - 1

ER -