Multiples of hypercyclic operators

Abstract

We give a negative answer to a question of Prăjitură by showing that there exists an invertible bilateral weighted shift T T on ℓ 2 ( Z ) \ell _2(\mathbb {Z}) such that T T and 3 T 3T are hypercyclic but 2 T 2T is not. Moreover, any G δ G_\delta set M ⊆ ( 0 , ∞ ) M \subseteq (0,\infty ) which is bounded and bounded away from zero can be realized as M = { t > 0 ∣ t T  is hypercyclic } M=\{t>0 \mid tT \textrm { is hypercyclic}\} for some invertible operator T T acting on a Hilbert space.