Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions
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| Format: | Preprint |
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2024
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| _version_ | 1866916222545690624 |
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| author | Hekkelman, Eva-Maria McDonald, Edward van Nuland, Teun D. H. |
| author_facet | Hekkelman, Eva-Maria McDonald, Edward van Nuland, Teun D. H. |
| contents | We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular $s$-summable spectral triples $(\mathcal{A}, \mathcal{H}, D)$, and an asymptotic expansion of $\mathrm{Tr}(P e^{-t(D+V)^2})$ as $t \downarrow 0$, where $P$ and $V$ belong to the algebra generated by $\mathcal{A}$ and $D$, and $V$ is bounded and self-adjoint. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_16338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions Hekkelman, Eva-Maria McDonald, Edward van Nuland, Teun D. H. Functional Analysis Mathematical Physics Operator Algebras Spectral Theory 47A55, 47A60, 47G30, 58B34, 47F10 We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular $s$-summable spectral triples $(\mathcal{A}, \mathcal{H}, D)$, and an asymptotic expansion of $\mathrm{Tr}(P e^{-t(D+V)^2})$ as $t \downarrow 0$, where $P$ and $V$ belong to the algebra generated by $\mathcal{A}$ and $D$, and $V$ is bounded and self-adjoint. |
| title | Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions |
| topic | Functional Analysis Mathematical Physics Operator Algebras Spectral Theory 47A55, 47A60, 47G30, 58B34, 47F10 |
| url | https://arxiv.org/abs/2404.16338 |