Speaker
Description
Let $\mathbb{D}$ be the unit disk and $\beta=(\beta_n)_{n\geq 0}$ a sequence of positive numbers satisfying $$\liminf_{n\to \infty} \beta_{n}^{1/n}\geq 1.$$ The associated Hardy space $H=H^{2}(\beta)\subset \mathcal{H}(\mathbb{D})$ is the set of analytic functions $f(z)=\sum_{n=0}^\infty a_n z^n$ such that $$\Vert f\Vert^2=\sum_{n=0}^\infty|a_n|^2 \beta_n<\infty.$$ Such are the Hardy, Bergman, Dirichlet, spaces ($\beta_n= 1,\ 1/(n+1), \ n+1$ respectively). In this talk, we will investigate sufficient, or necessary, conditions, on $\beta$ for all composition operators $C_\varphi,\ C_{\varphi}(f)=f\circ \varphi$, to be bounded on $H$. Here, $\varphi:\mathbb{D} \to \mathbb{D}$ is analytic. We will provide a simple necessary and sufficient condition when $\beta$ is (essentially) decreasing, meaning that $$\sup_{m\geq n} \frac{\beta_m}{\beta_n} \leq C<\infty.$$ This is joint work with P.Lef`evre, D.Li, L.Rodr\'iguez-Piazza.
