On approximation numbers of composition operators

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces Bα 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. Mathematics Subject Classification. Primary: 47B06 – Secondary: 47B33; 47B10 Key-words. approximation number – Bergman space – Carleson measure – composition operator – Hardy space – interpolation sequence – reproducing kernel – weighted Bergman space – weighted shift