Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk

Li, Daniel; Queffélec, Hervé; Rodríguez-Piazza, Luis

doi:10.1090/tran/7692

Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk
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by Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza PDF

Trans. Amer. Math. Soc. 372 (2019), 2631-2658 Request permission

Abstract:

We give examples of composition operators $C_\Phi$ on $H^2 ({\mathbb D}^2)$ 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–Ampère capacity of the image of $\Phi$.

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