Complete systems of unitary invariants for some classes of 2-isometries

Abstract

We characterize the unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition. This relies on a model for such operators built on operator-valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator-valued unilateral weighted shifts in a fairly general context. We also provide a complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition. This is formulated purely in the language of graph theory—namely, in terms of certain generation branching degrees. Finally, we study the membership of the Cauchy dual operators of $2$-isometries in classes $C_{0\cdot }$ and $C_{\cdot 0}$.