The structure of \({\mathcal {U}}_n\)-twisted power partial isometries

Abstract

Let \(n>1\) and let \(\{U_{ij}\}_{1\le i<j\le n}\) be \(n\atopwithdelims ()2\) commuting unitaries on a Hilbert space \({\mathcal {H}}\). Suppose \(U_{ji}:=U^*_{ij}\), \(1\le i<j\le n\). An n-tuple of power partial isometries \((V_1,...,V_n)\) on Hilbert space \({\mathcal {H}}\) is called \({\mathcal {U}}_n\)-twisted power partial isometry with respect to \(\{U_{ij}\}_{i<j}\) (or simply \({\mathcal {U}}_n\)-twisted power partial isometry if \(\{U_{ij}\}_{i<j}\) is clear from the context) if \(V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text {and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text {and}~i\ne j).\) We prove that each \({\mathcal {U}}_n\)-twisted power partial isometry admits a Halmos and Wallen (J Math Mech 19:657–663, 1969/1970) type orthogonal decomposition. We provide a concrete model for the decomposition of \({\mathcal {U}}_n\)-twisted power partial isometries.