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

Let $n>1$, and $\{U_{ij}\}$ for $1\le i<j\le n$ be $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ 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 $$T_i{T}_j=U_{ij}{T}_j{T}_i;T_i^{\ast }{T}_j=U_{ij}^{\ast }{T}_j{T}_i^{\ast }and{T}_k{U}_{ij}=U_{ij}{T}_k$$ for all $i,j,k=1,\dots ,n$ and $i\mathrm{\ne }j$. 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.