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

Abstract

We give examples of composition operators $C\_\hat{I}¦$ on $H^2 (\D^2)$ showing that the condition $\Vert \hat{I}¦\Vert \_\infty =1$ is not sufficient for their approximation numbers $a\_n(C\_\hat{I}¦)$ to satisfy $\lim \_n\to \infty [a\_n(C\_\hat{I}¦){]}^{1/\sqrt{n}}=1$, 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 $\hat{I}¦$.