Apparaît dans la collection : Composition operators and Banach spaces theory / Opérateurs de composition et espaces de Banach

In this talk we present some recent progress on the computation of the norm of composition operators acting on the Bloch space, which consists of analytic functions in the unit disk endowed with the norm $\quad|f|_{\mathcal{B}}=|f(0)|+\sup _{z \in \mathbb{D}}\left(1-|z|^2\right)\left|f^{\prime}(z)\right|$. The main approach relies on the hyperbolic geometry of the unit disk. We show that the hyperbolic derivative of the symbol, together with the hyperbolic distance between 0 and $\varphi(0)$, provides an upper bound for the norm of the composition operator. These geometric quantities offer a natural framework for controlling the norm and lead to useful estimates and, in many cases, exact norm formulas. Several representative examples - including affine symbols and Blaschke products - are discussed, along with conditions under which the obtained bounds are sharp.