Two results on composition operators on the Dirichlet space

We show that the decay of approximation numbers of compact composition operators on the Dirichlet space DD can be as slow as we wish. We also prove the optimality of a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi on boundedness on DD of self-maps of the disk all of whose powers are norm-bounded in DD.