A spectral radius type formula for approximation numbers of composition operators

For approximation numbers an(Cφ)an(Cφ) of composition operators CφCφ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that limn→∞⁡[an(Cφ)]1/n=e−1/Cap[φ(D)]limn→∞⁡[an(Cφ)]1/n=e−1/Cap[φ(D)], where Cap[φ(D)]Cap[φ(D)] is the Green capacity of φ(D)φ(D) in DD. This formula holds also for HpHp with 1≤p<∞1≤p<∞.