No abstract

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces B α 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.

Abstract. We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if f (s) = ∞ n=1 a n n −s with f ∞ := sup ℜs>0 |f (s)| < ∞, then ∞ n=1 |a n |n −2 ≤ f ∞ and even slightly better, and ∞ n=1 |a n |n −1/2 ≤ C f ∞ , C being an absolute constant.