Bounded perturbations of the Heisenberg commutation relation via dilation theory
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909579222188032 |
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| author | Gerhold, Malte Shalit, Orr |
| author_facet | Gerhold, Malte Shalit, Orr |
| contents | We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_07388 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounded perturbations of the Heisenberg commutation relation via dilation theory Gerhold, Malte Shalit, Orr Functional Analysis Operator Algebras 47A20, 47D03 We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows. |
| title | Bounded perturbations of the Heisenberg commutation relation via dilation theory |
| topic | Functional Analysis Operator Algebras 47A20, 47D03 |
| url | https://arxiv.org/abs/2208.07388 |