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

Palak Arora, Kelly Bickel, Constanze Liaw, Alan Sola

TL;DR

This work extends Clark theory to commuting pairs of compressed shifts on bidisk model spaces $K_\phi$, linking unitary perturbations $U_\alpha^1,U_\alpha^2$ to two-variable Clark measures $\sigma_\alpha$ via the embedding $J_\alpha$. The authors compute the adjoint $J_\alpha^*$, establish intertwining with multiplication on $L^2(\sigma_\alpha)$, and derive explicit formulas for the Clark unitaries under natural hypotheses, including rational inner functions. A central result is that the Taylor joint spectrum of $(U_\alpha^1,U_\alpha^2)$ coincides with the $\alpha$-level set $\mathcal{C}_\alpha=\{\zeta: \phi^*(\zeta)=\alpha\}$, i.e., with the support of $\sigma_\alpha$, for generic $\alpha$. The paper provides detailed examples (finite Blaschke products and a singular RIF) to illustrate when the theory yields explicit spectra and where pathologies arise, highlighting the rich interplay between function theory on the bidisk and multivariable operator theory.

Abstract

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 then use $J_α^*$ to obtain commuting unitaries on $K_φ$ which are often infinite-dimensional perturbations of compressed shift operators. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with $\mathrm{supp}(σ_α)\subset \mathbb{T}^2$ when $φ$ is a rational inner function.

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

TL;DR

This work extends Clark theory to commuting pairs of compressed shifts on bidisk model spaces , linking unitary perturbations to two-variable Clark measures via the embedding . The authors compute the adjoint , establish intertwining with multiplication on , and derive explicit formulas for the Clark unitaries under natural hypotheses, including rational inner functions. A central result is that the Taylor joint spectrum of coincides with the -level set , i.e., with the support of , for generic . The paper provides detailed examples (finite Blaschke products and a singular RIF) to illustrate when the theory yields explicit spectra and where pathologies arise, highlighting the rich interplay between function theory on the bidisk and multivariable operator theory.

Abstract

We develop a Clark theory for commuting compressed shift operators on model spaces associated with inner functions 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 as a weighted Cauchy transform of the Clark measure . Under natural assumptions, which generically include the case when is rational inner, we then use to obtain commuting unitaries on which are often infinite-dimensional perturbations of compressed shift operators. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with when is a rational inner function.

Paper Structure

This paper contains 13 sections, 16 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction and Overview
  2. Introduction
  3. Overview
  4. A Simple Example
  5. Preliminaries
  6. Clark Theory in One Variable
  7. Clark Measures for Two-Variable Inner Functions
  8. Taylor Joint Spectrum
  9. Clark Unitary Operators $(U_\alpha^1, U_\alpha^2)$
  10. RIF Level Sets and Taylor Joint Spectrum
  11. Examples
  12. Finite Blaschke Products
  13. A Singular Rational Inner Function

Key Result

Theorem 3.1

For each $\alpha$, the operator $U_{\alpha}$ acting on the model space $K_{\phi}$ is unitarily equivalent under $J_{\alpha}$ to the operator $M_{\zeta}$ on the spectral representation $L^2(\sigma_{\alpha})$.

Figures (2)

  • Figure 1: Support sets in $\mathbb{T}^2\simeq [-\pi, \pi)^2$ for the Clark measures $\sigma_{\alpha}$ associated with $\phi=z_1z_2$ and $\alpha=1$ (black), $\alpha=e^{i\frac{\pi}{4}}$ (gray), $\alpha=e^{i\frac{\pi}{2}}$ (orange), and $\alpha=e^{-i\frac{\pi}{4}}$ (pink).
  • Figure 2: Support sets for $\sigma_{\alpha}$ for $\phi=\frac{2z_1z_2-z_1-z_2}{2-z_1-z_2}$ and $\alpha=1$ (black), $\alpha=e^{i\frac{\pi}{4}}$ (gray), $\alpha=e^{i\frac{\pi}{2}}$ (orange), $\alpha=e^{-i\frac{\pi}{2}}$ (pink), and the exceptional value $\alpha=-1$ (red). Singular point $(1,1)\simeq (0,0)$ marked in red.

Theorems & Definitions (37)

  • Theorem 3.1
  • Example 3.2
  • Theorem 3.3: BickelSolaII, Theorem 3.3
  • Theorem 3.4: BickelSolaII, Theorem 4.2
  • Corollary 3.5
  • Proposition 3.6
  • Example 3.7: CT, Theorem 5
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • ...and 27 more