Details
Original language | English |
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Article number | 110477 |
Journal | Journal of Functional Analysis |
Volume | 287 |
Issue number | 4 |
Early online date | 24 Apr 2024 |
Publication status | Published - 15 Aug 2024 |
Abstract
Keywords
- math.OA, math.FA, 58J40, 58J42
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In: Journal of Functional Analysis, Vol. 287, No. 4, 110477, 15.08.2024.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on Rn
AU - Savin, Anton
AU - Schrohe, Elmar
PY - 2024/8/15
Y1 - 2024/8/15
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.
KW - math.OA
KW - math.FA
KW - 58J40, 58J42
U2 - 10.48550/arXiv.2303.14171
DO - 10.48550/arXiv.2303.14171
M3 - Article
VL - 287
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 4
M1 - 110477
ER -