On approximation numbers of composition operators

We show that the approximation numbers of a compact composition operator on the Hardy space H2H2 or on the weighted Bergman spaces BαBα of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they cannot decay more rapidly than exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bound and explicit an example.