ON CYCLICITY IN THE SPACE $H^{p}(\beta)$

Abstract

Let $\{\beta(n)\}$ be a sequence of positive numbers with $\beta(0) = 1$ and let $p\gt 0$. By the space $H^{p}(\beta)$, we mean the set of all formal power series $\sum^{\infty}_{n=0} \hat{f}(n) z^{n}$ for which $\sum^{\infty}_{n=0} |\hat{f}(n)|^{p} \beta(n)^{p} \lt \infty$. In this paper, we study cyclic vectors for the forward shift operator and supercyclic vectors for the backward shift operator on the space $H^{p} (\beta)$.