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 $f(z)=\sum_{n=0}^\infty c_n z^n$ such that

$$\Vert f\Vert^2=\sum_{n=0}^\infty|c_n|^2 \beta_n<\infty.$$

Such are the Hardy, Bergman, Dirichlet, spaces ($\beta_n= 1,\ 1/(n+1), \ n+1$ respectively).

For $\varphi:\mathbb{D} \to \mathbb{D}$ analytic, the composition operator $C_\varphi:H\to \mathcal{H}(\mathbb{D})$ with symbol $\varphi$ is defined by

$$C_{\varphi}(f)=f\circ \varphi.$$

In this talk, we will investigate conditions, on $\beta$ for {\bf all } composition operators $C_\varphi$ to be bounded on $H$. We will provide a simple necessary and sufficient condition when $\beta$ is essentially decreasing, meaning

$$\sup_{m\geq n} \frac{\beta_m}{\beta_n} \leq C<\infty.$$

We will also touch a problem on conditional multipliers.

This is joint work with P. Lefevre, D. Li, L. Rodriguez-Piazza.