The canonical Wold decomposition of commuting isometries with finite dimensional wandering spaces (Q385942)

The authors consider two commuting isometries \(V_1,V_2\) on a Hilbert space such that \(\text{dim\,ker}\,V^*_2\) is finite. They show that the classical Wold decomposition for \(V_1\) also reduces \(V_2\). From this result, the authors derive a similar assertion concerning an \(n\)-tuple of commuting isometries \(V_1,\dots,V_n\) such that \(\text{dim\, ker}\,V_1^*\dots V_n^*\) is finite.