Two remarks on composition operators on the Dirichlet space

TL;DR

This paper investigates the decay rates of approximation numbers for compact composition operators on the Dirichlet space and confirms the optimality of a boundedness criterion for certain self-maps of the disk.

Contribution

It demonstrates that the decay of approximation numbers can be arbitrarily slow and establishes the optimality of a boundedness condition for self-maps on the Dirichlet space.

Findings

Decay of approximation numbers can be arbitrarily slow

Boundedness of self-maps with all powers bounded is optimal

Provides new insights into composition operators on the Dirichlet space

Abstract

We show that the decay of approximation numbers of compact composition operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which was left open in the cited work. We also prove the optimality of a result of O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on $\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in $\mathcal{D}$.