The noncommutative residue or Wodzicki residue is (up to a constant multiple) the only nontrivial trace functional on the algebra of pseudodifferential operators (of arbitrary order) on a compact smooth manifold . For a it can be computed as the coefficient in front of in the asymptotic expansion of , where is the Laplacian, or equivalently, in terms of the usual residue, .
If is Riemannian, an elliptic operator of the negative integer order equal then, by a result of Connes, the Wodzicki residue coincides with the Dixmier trace?.
Mariusz Wodzicki, Spectral asymmetry and noncommutative residue, PhD thesis, Steklov Institute (1984)
Mariusz Wodzicki, Noncommutative residue. I. Fundamentals, in: K-theory, arithmetic and geometry, Lecture Notes in Math. 1289, Springer (1987) 320-399 [doi:10.1007%2FBFb0078372, MR923140]
wikipedia noncommutative residue
Nigel Kalton, Steven Lord, Denis Potapov, Fedor Sukochev, Traces of compact operators and the noncommutative residue, Adv. Math. 235 (2013) 1-55, arxiv/1210.3423 doi
Alain Connes, ; Noncommutative geometry, Academic Press 1984.
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