An extremal composition operator on the Hardy space of the bidisk with small approximation numbers

Abstract

We construct an analytic self-map $\hat{I}¦$ of the bidisk ${\mathbb{D}}^2$ whose image touches the distinguished boundary, but whose approximation numbers of the associated composition operator on $H^2({\mathbb{D}}^2)$ are small in the sense that ${\mathrm{limsup}}_{n\to \infty }[a_{n^2}(C_{\hat{I}}¦){]}^{1/n}<1$.