Approximation numbers of composition operators on the Dirichlet space

TL;DR

This paper investigates how quickly the approximation numbers of compact composition operators on the Dirichlet space decay, providing bounds and extending previous results to specific contact points and decay rates.

Contribution

It improves bounds on approximation numbers for these operators, focusing on contact points at 1 and demonstrating arbitrarily sub-exponentially small decay.

Findings

Upper and lower bounds for approximation numbers established

Contact points limited to the point 1 on the unit circle

Approximation numbers can decay arbitrarily sub-exponentially

Abstract

We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of O. El-Fallah, K. Kellay, M. Shabankhah and A. Youssfi, on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space D. We extend their results in two directions: first, the contact only takes place at the point 1. Moreover, the approximation numbers of the operator can be arbitrarily sub-exponentially small.