Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on $\mathbb R^n$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916262470221824 |
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| author | Savin, Anton Schrohe, Elmar |
| author_facet | Savin, Anton Schrohe, Elmar |
| contents | 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 $\mathop{\mathrm {Tr}}(D(H-λ)^{-K})$ as $|λ| \to \infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14171 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on $\mathbb R^n$ Savin, Anton Schrohe, Elmar Operator Algebras Functional Analysis 58J40, 58J42 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 $\mathop{\mathrm {Tr}}(D(H-λ)^{-K})$ as $|λ| \to \infty$ 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. |
| title | Noncommutative Residues, Equivariant Traces, and Trace Expansions for an Operator Algebra on $\mathbb R^n$ |
| topic | Operator Algebras Functional Analysis 58J40, 58J42 |
| url | https://arxiv.org/abs/2303.14171 |