Composition operators and Rational Inner Functions on the bidisc

TL;DR

The paper analyzes composition operators $C_{\Phi}$ induced by rational inner function coordinates on the bidisc $\mathbb{D}^2$, acting on weighted Bergman spaces $A^2_\beta(\mathbb{D}^2)$. It shows unboundedness on the classical Bergman space $A^2(\mathbb{D}^2)$ under a one-singularity boundary condition with a quantitative lower bound on the denominator, while proving boundedness from $A^2_{\frac{\beta}{2}-2}(\mathbb{D}^2)$ to $A^2_{\beta}(\mathbb{D}^2)$ for $\beta>4$ when the RIF is induced by a stable polynomial. A third result provides a general volume-condition in the Schur-Agler class that ensures boundedness on $A^2_\beta(\mathbb{D}^n)$, connecting Agler decompositions to operator-boundedness via Carleson-type measures. The work combines Łojasiewicz-type inequalities, Carleson-volume estimates, and Agler decompositions to extend the theory of composition operators to singular RIF symbols on the bidisc and offers guidance for higher-dimensional polydiscs.

Abstract

In the present article, composition operators induced by Rational Inner Functions on the bidisc $\mathbb{D}^2$ are studied, acting on the weighted Bergman space $A^2_β(\mathbb{D}^2).$ We prove that under mild conditions that Rational Inner Functions with one singularity on $\mathbb{T}^2$ induce unbounded composition operator on $A^2(\mathbb{D}^2).$ We also prove that under the condition of stability of the polynomial inducing the Rational Inner Function, the composition operator is bounded between two different Bergman spaces.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Example 3.5
• Theorem 3.6