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

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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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Dates and versions

hal-01536919 , version 1 (12-06-2017)
hal-01536919 , version 2 (01-03-2018)

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Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza. Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk. 2018. ⟨hal-01536919v2⟩
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