Cyclicity of bicyclic operators and completeness of translates

Abstract

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if S : (a n ) n∈Z −→ (a n−1 ) n∈Z is the shift operator acting on the weighted space of sequences 2 ω (Z), if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if lim n→+∞ log ω(−n)/ √ n = 0. On the other hand one can see that S is not cyclic if the series n≥1 log ω(−n)/n 2 diverges. We show that the question of Herrero whether either S or S * is cyclic on 2 ω (Z) admits a positive answer when the series n∈Z log ||S n ||/(n 2 + 1) is convergent. We also prove completeness results for translates in certain Banach spaces of functions on R.123 294 E. Abakumov et al.

A weighted bilateral shift with cyclic square is supercyclic

“…We would like to stress that the following problem remains open. As mentioned in [1], it is not known whether certain specific weighted bilateral shifts are cyclic. For instance, let 0 < α 1, w n = 1 if n 1 and w n = 1 − n −α if n 2.…”

“…Finally, we shall prove yet another sufficient condition for cyclicity of a weighted bilateral shift. It does not follow from any known sufficient condition including the following most recent one due to Abakumov, Atzmon and Grivaux [1].…”

“…Then the weighted bilateral shift T = T w,p is cyclic if and only if the sequence {α −1 n } n∈Z does not belong to ℓ q (Z), where 1 p + 1 q = 1. This highly non-trivial result does not give a characterization of cyclicity for weighted bilateral shifts.…”

“…Let w = {w n } n∈Z be a bounded sequence of non-zero complex numbers. Then T w,p is supercyclic if and only if T w,p ⊕ T w,p ′ is cyclic, where 1 p + 1 p ′ = 1. Proof.…”

Cyclicity in ℓ spaces and zero sets of the Fourier transforms

“…Since K is a Helson set, we consider δ 2 the constant satisfying the condition (iii) of Definition 3.3. Since the set of cyclic vectors in I({1}) is dense in I({1}) (see [1]), there exists a function h cyclic in I({1}) such that…”