Hypercyclic Weighted Shifts

Abstract

Abstract. An operator T acting on a Hilbert space is hypercyclic if, for some vector x in the space, the orbit {T"x: n > 0} is dense. In this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic. As a consequence, we show within the class of weighted shifts that multi-hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic. For general hypercyclic operators the corresponding questions were posed by D. A. Herrero, and they still remain open. Using a different technique we prove that / + T is hypercyclic whenever T is a unilateral backward weighted shift, thus answering in more generality a question recently posed by

Hypercyclic operators with an in¯nite dimensional closed subspace of periodic points

“…Theorem 1.3 is related to a result of H. Salas, who proved in [11] that when (ω i ) i≥0 is any bounded sequence of positive scalars and S is the backward shift defined on X by Se 0 = 0 and for every i ≥ 0, Se i+1 = ω i e i , then the operator I + S is hypercyclic. Then the operator I +T , where I denotes the identity operator on X, is hypercyclic on X, and it is equal to the identity operator on the closed space spanned by the vectors e 2i , i ≥ 0.…”

“…Let (z k ) k≥1 be a dense sequence of vectors of 2 with finite support such that for every k ≥ 1, max supp(z k ) ≤ k. Proceeding as in the proof of Theorem 3.3 in [11], we will construct inductively a fast increasing sequence (n k ) k≥1 of integers and a sequence (y k ) k≥1 of finitely supported vectors of 2 such that:…”

Disjoint mixing operators

“…It is a much stronger form of a lemma by Salas [24] that he used to prove that any perturbation of the identity by adding a backward weighted shift on 1 is hypercyclic. To be more precise, we obtain a multi-approximation version of Salas' lemma, with fine estimates.…”

J-Class Weighted Shifts on the Space of Bounded Sequences of Complex Numbers

“…Salas [10] was the first who characterized the hypercyclic weighted shifts in terms of their weight sequences. We would like to point out that l ∞ (N) and l ∞ (Z) do not support hypercyclic operators since they are not separable Banach spaces.…”