Nuclearity and Grothendieck-Lidskii formula for quaternionic operators
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909119058804736 |
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| author | Cerejeiras, Paula Colombo, Fabrizio Pinos, Alberto Debernardi Kähler, Uwe Sabadini, Irene |
| author_facet | Cerejeiras, Paula Colombo, Fabrizio Pinos, Alberto Debernardi Kähler, Uwe Sabadini, Irene |
| contents | We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14189 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nuclearity and Grothendieck-Lidskii formula for quaternionic operators Cerejeiras, Paula Colombo, Fabrizio Pinos, Alberto Debernardi Kähler, Uwe Sabadini, Irene Classical Analysis and ODEs Functional Analysis 46S05, 47S05 (Primary) 47B10 (Secondary) We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace. |
| title | Nuclearity and Grothendieck-Lidskii formula for quaternionic operators |
| topic | Classical Analysis and ODEs Functional Analysis 46S05, 47S05 (Primary) 47B10 (Secondary) |
| url | https://arxiv.org/abs/2306.14189 |