Cyclicity in ℓ spaces and zero sets of the Fourier transforms

Abstract

Abstract. We study the cyclicity of vectors u in ℓ p (Z). It is known that a vector u is cyclic in ℓ 2 (Z) if and only if the zero set, Z( u), of its Fourier transform, u, has Lebesgue measure zero and log | u| ∈ L 1 (T), where T is the unit circle. Here we show that, unlike ℓ 2 (Z), there is no characterization of the cyclicity of u in ℓ p (Z), 1 < p < 2, in terms of Z( u) and the divergence of the integral T log | u|. Moreover we give both necessary conditions and sufficient conditions for u to be cyclic in ℓ p (Z), 1 < p < 2.