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

TL;DR

This paper extends Clark theory to multivariable settings on the bidisk, analyzing joint spectra of certain unitary operators related to inner functions, with implications for operator theory and spectral analysis.

Contribution

It develops a Clark theory for commuting operators on the bidisk, identifying their joint spectra and establishing unitary equivalences, which advances understanding of multivariable operator models.

Findings

Clark unitaries are unitarily equivalent to multiplication operators

Taylor joint spectrum matches level sets of the inner function

Results include cases where the inner function is rational inner

Abstract

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 $J_{\alpha} \colon K_{\phi}\to L^2(\sigma_{\alpha})$ 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.