Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\phi }$ associated with inner functions $\phi$ on the bidisk, which exhibits both similarities and marked differences compared to the classical one-variable version. We first identify the adjoint of the embedding operator $J_{\alpha }\text{\hspace{0.17em}⁣}:K_{\phi }\to L^2({\sigma }_{\alpha })$ as a weighted Cauchy transform of the Clark measure ${\sigma }_{\alpha }$. Under natural assumptions, which generically include the case when $\phi$ is rational inner, we obtain commuting unitaries on $K_{\phi }$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_{\phi }$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2({\sigma }_{\alpha })$ and then establish a number of related properties and simplified results in special cases. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with level sets of $\phi$ when $\phi$ is a rational inner function.