$q$-commuting dilation

Abstract

In this paper, we prove that any pair of $q$-commuting contractions on a Hilbert space dilates to a pair of $q$-commuting unitaries, where $|q|=1$. We generalize this result to a $(G,\mathbf {q})$-commuting $n$-tuple $(T_1,\ldots ,T_n)$ of strict contractions, where $G$ is an acyclic graph with vertex set $\{1,\ldots ,n\}$. We further generalize it to a family of $(G,\mathbf {q})$-commuting strict contractions, where $G$ is an acyclic graph on an infinite set of vertices.