Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc

Mouze, Augustin ; Munnier, Vincent

Canadian mathematical bulletin, Tome 64 (2021) no. 2, pp. 264-281

Résumé

For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$ $$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$$We establish the optimal growth of frequently hypercyclic functions for$T_\alpha $ in terms of $L^p$ averages, $1\leq p\leq +\infty $. This allows us to highlight a critical exponent.

Détail

DOI : 10.4153/S0008439520000430

Mots-clés : Frequently hypercyclic operator, Taylor shift, boundary behavior