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A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry

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Abstract

We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman–Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes’ notation for quantised calculus, we prove that for a wide class of p-summable spectral triples \(({\mathcal {A}},H,D)\) and self-adjoint \(V \in {\mathcal {A}}\), there holds

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Notes

  1. In particular, \(V^r \in {\mathcal {L}}_1\) for every \(r>p.\)

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16801, USA

    E. McDonald

  2. School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

    F. Sukochev & D. Zanin

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McDonald, E., Sukochev, F. & Zanin, D. A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry. Lett Math Phys 112, 77 (2022). https://doi.org/10.1007/s11005-022-01568-5

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