Wold-type decomposition for ${\mathcal{U}}_n$-twisted contractions

Abstract

Let $n>1$, and $\{U_{ij}\}$ for $1\le i<j\le n$ be $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ commuting unitaries on a Hilbert space $\mathcal{H}$ such that $U_{ji}:=U_{ij}^{\ast }$. An $n$-tuple of contractions $(T_1,\dots ,T_n)$ on $\mathcal{H}$ is called ${\mathcal{U}}_n$-twisted contraction with respect to a twist $\{U_{ij}{\}}_{i<j}$ if $T_1,\dots ,T_n$ satisfy \[ T_iT_j=U_{ij}T_jT_i; \hspace{0.5cm} \hspace{1cm} T_i^*T_j= U^*_{ij}T_jT_i^* \hspace{0.5cm} \mbox{and} \hspace{0.5cm} T_kU_{ij} =U_{ij}T_k \] for all $i,j,k=1,\dots ,n$ and $i\ne j$. We obtain a recipe to calculate the orthogonal spaces of the Wold-type decomposition for ${\mathcal{U}}_n$-twisted contractions on Hilbert spaces. As a by-product, a new proof as well as complete structure for ${\mathcal{U}}_2$-twisted (or pair of doubly twisted) and ${\mathcal{U}}_n$-twisted isometries have been established.

arXiv comment20 pages, title changed and revised. To appear in Journal of Mathematical Analysis and Applications