# Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk

Abstract : We give examples of composition operators $C_\Phi$ on $H^2 (\D^2)$ showing that the condition $\|\Phi \|_\infty = 1$ is not sufficient for their approximation numbers $a_n (C_\Phi)$ to satisfy $\lim_{n \to \infty} [a_n (C_\Phi) ]^{1/\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Amp\`ere capacity of the image of $\Phi$.
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https://hal-univ-artois.archives-ouvertes.fr/hal-01536919
Contributor : Daniel Li <>
Submitted on : Monday, June 12, 2017 - 11:36:49 AM
Last modification on : Friday, November 27, 2020 - 2:18:02 PM
Long-term archiving on: : Thursday, September 14, 2017 - 12:41:07 PM

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Examples-bidisk.pdf
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### Identifiers

• HAL Id : hal-01536919, version 1
• ARXIV : 1706.03570

### Citation

Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza. Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk. 2017. ⟨hal-01536919v1⟩

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