On approximation numbers of composition operators

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces $\mathfrak{B}_\alpha$ of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example

Some new properties of composition operators associated with lens maps

We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space $H^2$. The last ones are connected with Hardy-Orlicz and Bergman-Orlicz spaces $H^\psi$ and $B^\psi$, and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact.Comment: 21 page