Composition operators and Rational Inner Functions on the bidisc

TL;DR

This paper investigates the properties of composition operators induced by Rational Inner Functions on the bidisc, focusing on their boundedness and unboundedness on weighted Bergman spaces.

Contribution

It establishes conditions under which Rational Inner Functions induce bounded or unbounded composition operators on Bergman spaces of the bidisc.

Findings

Rational Inner Functions with one singularity induce unbounded operators.

Stable inducing polynomials lead to bounded composition operators between different Bergman spaces.

Conditions for boundedness depend on the singularity and stability of the inducing polynomial.

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 $A^2_{\beta}(\mathbb{D}^2).$ 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.