Compatible pairs of commuting isometries (Q2348021)

The paper under review studies the structure of pairs of commuting isometries. Two commuting isometries \(V_1\) and \(V_2\) are called \textit{compatible} if the orthogonal projections onto the ranges of \(V_1^{m}\) and \(V_2^n\) commute for all non-negative integers \(m\) and \(n\). Two commuting isometries are \textit{completely non-compatible} if there is no reducing subspaces on which they are compatible. It is shown that any pair of commuting isometries decomposes as a direct sum of a compatible pair and a completely non-compatible pair. The paper focuses on describing the compatible part of such a decomposition. The authors make use of the notion of \textit{diagrams} in their investigation. A subset of \(\mathbb{Z}^2\) is called a diagram if it is invariant under all translation by vectors with non-negative integer components. Each diagram together with a given Hilbert space gives rise to a compatible pair of commuting isometries. However, there are compatible pairs not given by such a fashion and the authors introduce the notion of \textit{pairs of generalized powers} to handle those cases. Several results are obtain. The most general result (Theorem 8.3) states the following: any non-doubly commuting, compatible pair of commuting isometries can be decomposed into a sum of isometries such that each summand is defined by a diagram or is a pair of generalized powers. Examples are given throughout the paper.