RHALY OPERATORS ACTING ON \(\ell ^p\)-SPACES

Let \(\mathcal S\) $\mathcal{S}$ be the space of all complex sequences. If \(\{ \beta _n\}_{n=0}^\infty \in \mathcal S\) ${\{{\beta }_n\}}_{n=0}^{\infty }\in \mathcal{S}$ , the Rhaly operator \(R_{\{ \beta _n\} }\) $R_{\{{\beta }_n\}}$ is defined in \(\mathcal {S}\) $\mathcal{S}$ as follows: If \(\{ a_n\}_{n=0}^\infty \in \mathcal S\) ${\{a_n\}}_{n=0}^{\infty }\in \mathcal{S}$ , then \(R_{ \{\beta _n\} }(\{ a_n\}_{n=0}^\infty )\,=\,\left\{ \beta _n\sum _{k=0}^na_k\right\} _{n=0}^\infty .\) $$R_{\{{\beta }_n\}}({\{a_n\}}_{n=0}^{\infty })={\left\{{\beta }_n,\sum _{k=0}^n,a_k\right\}}_{n=0}^{\infty }.$$ Rhaly operators are a natural generalization of the Cesàro operator. We characterize the sequences \(\{ \beta _n\}_{n=0}^\infty\) ${\{{\beta }_n\}}_{n=0}^{\infty }$ for which the operator \(R_{\{ \beta _n\} }\) $R_{\{{\beta }_n\}}$ is either bounded or compact on the space \(\ell ^p\) ${\ell }^p$ for any \(p\in (1, \infty )\) $p\in (1,\infty )$ .