Composition operators on Banach spaces of formal power series

Yousefi, B. ; Jahedi, S.

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 481-487

Résumé

Let $\{\beta (n)\}^{\infty}_{n=0}$ be a sequence of positive numbers and $1\leq p \infty$. We consider the space $H^{p}(\beta)$ of all power series $f(z)= \sum_{n=0}^{\infty} \hat{f}(n)z^{n}$ such that $\sum_{n=0}^{\infty}|\hat{f}(n)|^{p}\beta(n)^{p}\infty $ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and $\sum_{n=1}^{\infty} \frac{n^{qj}}{\beta(n)^{q}}=\infty$ for some nonnegative integer $j$. We show that if $C_{\varphi}$ is compact on $H^{p}(\beta)$, then the non-tangential limit of $\varphi^{(j+1)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\varphi}$ is Fredholm on $H_{p}(\beta)$, then $\varphi$ must be an automorphism of the open unit disc.