Bounded perturbations of the Heisenberg commutation relation via dilation theory
Abstract
We extend the notion of dilation distance to strongly continuous oneparameter 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 dtuples of oneparameter 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 higherdimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.07388
 Bibcode:
 2022arXiv220807388G
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras;
 47A20;
 47D03
 EPrint:
 9 pages