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

Abstract

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\varphi }$ associated with inner functions $\varphi$ 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 $J_{\alpha }:K_{\varphi }\to L^2({\sigma }_{\alpha })$ as a weighted Cauchy transform of the Clark measure ${\sigma }_{\alpha }$. Under natural assumptions, which generically include the case when $\varphi$ is rational inner, we obtain commuting unitaries on $K_{\varphi }$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_{\varphi }$. 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 $\varphi$ when $\varphi$ is a rational inner function.

32 pages, 2 figures. In this version we added several results, and refined some items such as: added Remark 4.8 and 4.9, refined Theorem 5.1 and the corresponding results, expanded Example 6.2