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

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 $V_1{V}_2=V_2\phi (V_1)$, where $\phi$ 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 $(M_z,M_{\mathrm{\Theta }}{W}_{\phi }^{\ast })$ acting on $H^2(\mathcal{F})$, where $W_{\phi }$ is a unitary weighted composition operator on the Hardy space $H^2$ and $\mathrm{\Theta }\in H^{\mathrm{\infty }}(\mathbb{D},\mathcal{B}(\mathcal{F}))$ is an operator-valued inner function such that $M_{\mathrm{\Theta }}$ 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 $\mathrm{\mid }q\mathrm{\mid }=1$).