## Abstract

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 Rørdam [Duke Math. J. 77 (1995), pp. 627–656] 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 Rørdam’s result, and in typical cases we significantly improve control of the bound when the dimension grows.

Original language | English |
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Pages (from-to) | 3949-3957 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 151 |

Issue number | 9 |

DOIs | |

State | Published - 1 Sep 2023 |

## All Science Journal Classification (ASJC) codes

- Applied Mathematics
- General Mathematics