Composition operators and Rational Inner Functions on the bidisc: A geometric approach

TL;DR

This paper provides a geometric criterion for the boundedness of composition operators induced by rational inner functions on weighted Bergman spaces of the bidisc, extending previous characterizations to non-smooth cases.

Contribution

It introduces a uniform geometric criterion for boundedness of composition operators induced by RIFs on the bidisc, including non-smooth symbols, expanding prior results.

Findings

Boundedness is equivalent to transversal intersection of level sets for non-smooth symbols.

High-order tangential intersections at singularities are permissible for boundedness.

The criterion extends Bayart-Kosiński's characterization to non-smooth self maps of the bidisc.

Abstract

We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_{\Phi}:A^2_{\beta}(\mathbb{D}^2)\to A^2_{\beta}(\mathbb{D}^2)$ where $\Phi$ is induced by rational inner functions (RIFs) or a RIF and a smooth function (mixed case). Our approach is geometric. Our main result is a uniform criterion for all $\beta\in(-1,0]$ that can be summarized as follows: Boundedness of the composition operator is equivalent to transversal intersection of the level sets for non-smooth symbols, under the assumption that if any tangential intersection occurs on the singularity it must be of high order. This extends the characterization of Bayart-Kosi\'nski to the non-smooth self maps of the bidisc. To reach our conclusions, we utilize results obtained by Anderson, Bergqvist, Bickel, Cima and Sola on Clark measures associated to RIFs and Puiseux factorizations.