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

TL;DR

This paper explores specific examples of composition operators on the Hardy space of the bi-disk, demonstrating that certain conditions on the symbol function do not guarantee expected decay rates of approximation numbers, unlike in one dimension.

Contribution

It provides counterexamples and conditions linking approximation numbers of composition operators on the bi-disk to Monge-Ampère capacity, extending understanding beyond the one-dimensional case.

Findings

Counterexamples where $ orm{\Phi}_\infty=1$ does not imply the decay condition.

A scenario where the decay condition holds.

Connection between approximation numbers and Monge-Ampère capacity.

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 $\Phi$.