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Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on Rn

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Anton Savin
  • Elmar Schrohe

Research Organisations

External Research Organisations

  • Peoples' Friendship University of Russia (RUDN)

Details

Original languageEnglish
Article number110477
JournalJournal of Functional Analysis
Volume287
Issue number4
Early online date24 Apr 2024
Publication statusPublished - 15 Aug 2024

Abstract

We consider an algebra \(\mathscr A\) of Fourier integral operators on \(\mathbb R^n\). It consists of all operators \(D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)\) on the Schwartz space \(\mathscr S(\mathbb R^n)\) that can be written as finite sums \(\) D= \sum R_gT_w A, \(\) with Shubin type pseudodifferential operators \(A\), Heisenberg-Weyl operators \(T_w\), \(w\in \mathbb C^n\), and lifts \(R_g\), \(g\in \mathrm U(n)\), of unitary matrices \(g\) on \(\mathbb C^n\) to operators \(R_g\) in the complex metaplectic group. For \(D \in \mathscr A\) and a suitable auxiliary Shubin pseudodifferential operator \(H\) we establish expansions for \(\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})\) as \(|\lambda| \to \infty\) in a sector of \(\mathbb C\) for sufficiently large \(K\) and of \(\mathop{\mathrm {Tr}}(De^{-tH})\) as \(t\to 0^+\). We also obtain the singularity structure of the meromorphic extension of \(z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})\) to \(\mathbb C\). Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

Keywords

    math.OA, math.FA, 58J40, 58J42

Cite this

Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on Rn. / Savin, Anton; Schrohe, Elmar.
In: Journal of Functional Analysis, Vol. 287, No. 4, 110477, 15.08.2024.

Research output: Contribution to journalArticleResearchpeer review

Savin A, Schrohe E. Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on Rn. Journal of Functional Analysis. 2024 Aug 15;287(4):110477. Epub 2024 Apr 24. doi: 10.48550/arXiv.2303.14171, 10.1016/j.jfa.2024.110477
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N2 - We consider an algebra \(\mathscr A\) of Fourier integral operators on \(\mathbb R^n\). It consists of all operators \(D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)\) on the Schwartz space \(\mathscr S(\mathbb R^n)\) that can be written as finite sums \(\) D= \sum R_gT_w A, \(\) with Shubin type pseudodifferential operators \(A\), Heisenberg-Weyl operators \(T_w\), \(w\in \mathbb C^n\), and lifts \(R_g\), \(g\in \mathrm U(n)\), of unitary matrices \(g\) on \(\mathbb C^n\) to operators \(R_g\) in the complex metaplectic group. For \(D \in \mathscr A\) and a suitable auxiliary Shubin pseudodifferential operator \(H\) we establish expansions for \(\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})\) as \(|\lambda| \to \infty\) in a sector of \(\mathbb C\) for sufficiently large \(K\) and of \(\mathop{\mathrm {Tr}}(De^{-tH})\) as \(t\to 0^+\). We also obtain the singularity structure of the meromorphic extension of \(z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})\) to \(\mathbb C\). Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

AB - We consider an algebra \(\mathscr A\) of Fourier integral operators on \(\mathbb R^n\). It consists of all operators \(D: \mathscr S(\mathbb R^n)\to \mathscr S(\mathbb R^n)\) on the Schwartz space \(\mathscr S(\mathbb R^n)\) that can be written as finite sums \(\) D= \sum R_gT_w A, \(\) with Shubin type pseudodifferential operators \(A\), Heisenberg-Weyl operators \(T_w\), \(w\in \mathbb C^n\), and lifts \(R_g\), \(g\in \mathrm U(n)\), of unitary matrices \(g\) on \(\mathbb C^n\) to operators \(R_g\) in the complex metaplectic group. For \(D \in \mathscr A\) and a suitable auxiliary Shubin pseudodifferential operator \(H\) we establish expansions for \(\mathop{\mathrm {Tr}}(D(H-\lambda)^{-K})\) as \(|\lambda| \to \infty\) in a sector of \(\mathbb C\) for sufficiently large \(K\) and of \(\mathop{\mathrm {Tr}}(De^{-tH})\) as \(t\to 0^+\). We also obtain the singularity structure of the meromorphic extension of \(z\mapsto \mathop{\mathrm{Tr}}(DH^{-z})\) to \(\mathbb C\). Moreover, we find a noncommutative residue as a suitable coefficient in these expansions and construct from it a family of localized equivariant traces on the algebra.

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