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

Abstract

We give examples of composition operators C Φ C_\Phi on H 2 ( D 2 ) H^2 ({\mathbb D}^2) showing that the condition ‖ Φ ‖ ∞ = 1 \|\Phi \|_\infty = 1 is not sufficient for their approximation numbers a n ( C Φ ) a_n (C_\Phi ) to satisfy lim n → ∞ [ a n ( C Φ ) ] 1 / n = 1 \lim _{n \to \infty } [a_n (C_\Phi ) ]^{1/\sqrt {n}} = 1 , contrary to the 1 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 Φ \Phi .