A note on non-supercyclic vectors of Hilbert space operators

Abstract

In this note it is shown that there is a bounded linear operator T on the Hardy Hilbert space H2 and a vector f in H2 such that the closure of the set {αTnf:α∈ℂ,n≥0} is not H2, but for every subsequence (nk)k=1∞ the closed linear span of {Tnkf:k≥1} is the whole space H2. Furthermore, the closure of {Tng:n≥0} is H2 for some g∈H2.