2025/11/07 by Arora, Palak, Bickel, Kelly, Liaw, Constanze +1
Mathematics · #47A13 #47A20 #47A55 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods
paper · doi:10.48550/arxiv.2511.05306
openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · openalex updated_date 2026/07/28
We develop a Clark theory for commuting compressed shift operators on model spaces Kϕ associated with inner functions ϕ 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 Jα \colon Kϕ→ L2(σα) as a weighted Cauchy transform of the Clark measure σα. Under natural assumptions, which generically include the case when ϕ is rational inner, we obtain commuting unitaries on Kϕ that are (often infinite-dimensional) perturbations of the compressed shift operators Kϕ. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on L2(σα) 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 ϕ when ϕ is a rational inner function.