Let \(T=C_\varphi,\ Tf=f\circ \varphi\) be a compact
composition operator on a Hilbert space \(H\) of analytic functions
on the unit disk \(\mathbb{D}\), and \(a_{n}(T)\) be its \(n\)-th
approximation (=singular) number. The decay rate of \(a_{n}(T)\) to
\(0\) according to the analytic self-map \(\varphi\colon
\mathbb{D}\to \mathbb{D}\) is only partially understood so far, and
moreover, for a given \(\varphi\), can vary much according to the
space \(H\) (Hardy, Dirichlet,\ldots). In this talk, we will yet see
a common feature to all of those spaces, under the form of an exact
formula
\[
\beta(T):=\lim_{n\to \infty} $\,$ [a_{n}(T)]^{1/n}=M(K)
\]
the existence of the limit being part of the conclusion. Here,
\[
M(K)=e^{-1/C(K)}
\]
where \(C(K)\) is the Green capacity of
\(K=\overline{\varphi(\mathbb{D})}\) in \(\mathbb{D}\). This
formula is somehow analog to the spectral radius
formula \[\lim_{n\to \infty}\Vert T^n\Vert^{1/n}=\sup_{\lambda\in
\sigma(T)}|\lambda|\] the image of the symbol and its capacity
replacing the spectrum of the operator and its supremum.
As an application, we recover in a simplified and unified way
several recent results on those composition operators, acting either
on the Hardy, Bergman, or Dirichlet spaces.
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