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

## TL;DR

This paper constructs a specific composition operator on the Hardy space of the bidisk with an extremal property, demonstrating that it can have small approximation numbers despite its image touching the boundary.

## Contribution

It provides a novel example of an extremal composition operator on the bidisk with small approximation numbers, highlighting new phenomena in operator theory.

## Key findings

- Approximation numbers of the constructed operator decay rapidly.
- The image of the map touches the boundary, yet the operator remains compact.
- The result challenges previous assumptions about boundary behavior and approximation numbers.

## Abstract

We construct an analytic self-map $\Phi$ 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 $\limsup_{n \to \infty} [a_{n^2} (C_\Phi)]^{1 / n} < 1$.