Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914568321630208 |
|---|---|
| author | Arora, Palak Bickel, Kelly Liaw, Constanze Sola, Alan |
| author_facet | Arora, Palak Bickel, Kelly Liaw, Constanze Sola, Alan |
| contents | We develop a Clark theory for commuting compressed shift operators on model spaces $K_ϕ$ 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 $J_α \colon K_ϕ\to L^2(σ_α)$ as a weighted Cauchy transform of the Clark measure $σ_α$. Under natural assumptions, which generically include the case when $ϕ$ is rational inner, we obtain commuting unitaries on $K_ϕ$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_ϕ$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2(σ_α)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra Arora, Palak Bickel, Kelly Liaw, Constanze Sola, Alan Complex Variables Functional Analysis 47A13, 47A20, 47A55 We develop a Clark theory for commuting compressed shift operators on model spaces $K_ϕ$ 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 $J_α \colon K_ϕ\to L^2(σ_α)$ as a weighted Cauchy transform of the Clark measure $σ_α$. Under natural assumptions, which generically include the case when $ϕ$ is rational inner, we obtain commuting unitaries on $K_ϕ$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_ϕ$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2(σ_α)$ 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. |
| title | Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra |
| topic | Complex Variables Functional Analysis 47A13, 47A20, 47A55 |
| url | https://arxiv.org/abs/2511.05306 |