We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{ Z})$. It is known that a vector $u$ is cyclic in $\ell^2(\mathbb{ Z})$ if and only if the zero set, $\mathcal{ Z}(\widehat{ u})$, of its Fourier transform, $\widehat{ u}$, has Lebesgue measure zero and $\log |\widehat{ u}| \not \in L^1(\mathbb{ T})$, where $\mathbb{ T}$ is the unit circle. Here we show that, unlike $\ell^2(\mathbb{ Z})$, there is no characterization of the cyclicity of $u$ in $\ell^p(\mathbb{ Z})$, $1<p<2$, in terms of $\mathcal{ Z}(\widehat{ u})$ and the divergence of the integral $\int\_\mathbb{ T} \log |\widehat{ u}| $. Moreover we give both necessary conditions and sufficient conditions for $u$ to be cyclic in $\ell^p(\mathbb{ Z})$, $1<p<2$.
Categories: math.FA math.CA