A note on supercyclic vectors of Hilbert space operators

Abstract

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists x∈H such that the closure of {αTnx:α∈C,n⩾0} is H, then there is a subsequence (nk)k=1∞ such that the closed linear span of {Tnkx:k⩾1} is not the whole space H.