Composition operators and Rational Inner Functions on the bidisc: A geometric approach
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917427946717184 |
|---|---|
| author | Beslikas, Athanasios |
| author_facet | Beslikas, Athanasios |
| contents | We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_Φ:A^2_β(\mathbb{D}^2)\to A^2_β(\mathbb{D}^2)$ where $Φ$ 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 $β\in(-1,0]$ 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-Kosiński 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20330 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Composition operators and Rational Inner Functions on the bidisc: A geometric approach Beslikas, Athanasios Complex Variables 32A08, 32A37 We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_Φ:A^2_β(\mathbb{D}^2)\to A^2_β(\mathbb{D}^2)$ where $Φ$ 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 $β\in(-1,0]$ 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-Kosiński 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. |
| title | Composition operators and Rational Inner Functions on the bidisc: A geometric approach |
| topic | Complex Variables 32A08, 32A37 |
| url | https://arxiv.org/abs/2604.20330 |