Functional Model for Isometric Pairs Satisfying an Automorphic Covariance Relation - I

Abstract

We derive a canonical functional model for a pair of isometric operators $(V_1, V_2)$ acting on a Hilbert space $\mathcal{H}$ that satisfy the automorphic covariance relation $V_1 V_2 = V_2 φ(V_1)$, where $φ$ is an automorphism of the unit disk $\mathbb{D}$. This extends the classical Bercovici--Douglas--Foias functional model for commuting isometric pairs to a dynamically twisted setting. When both $V_1$ and $V_2$ are pure isometries (shifts), the model simplifies to $(M_z, M_Θ W_φ^*)$ acting on $H^2(\mathcal{F})$, where $W_φ$ is a unitary weighted composition operator on the Hardy space $H^2$ and $Θ\in H^\infty(\mathbb{D}, \mathcal{B}(\mathcal{F}))$ is an operator-valued inner function such that $M_Θ$ is a shift. As an application, we establish a complete unitary classification theorem for automorphically covariant isometric pairs by identifying a complete set of joint unitary invariants. Notably, our results are novel even in the special case of rotation automorphisms ($V_1 V_2 = q V_2 V_1$ with $|q| = 1$).