A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ 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/Capφ(D)]}${\lim }_{n\to \infty }?{a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}\phi (\mathbb{D})]}$, where Capφ(D)]$\mathrm{Cap}\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.