Cyclic properties of Volterra operator

Abstract

A bounded linear operator T defined on a Hilbert space H is said to be supercyclic if there exists a vector x ∈ H such that the set {λT n x : n ∈ N, λ ∈ C} is dense in H. In the present work, two open questions posed by N. H. Salas and J. Zemánek respectively, are solved. Namely, we will exhibit that the classical Volterra operator V and the identity plus Volterra operator I + V are not supercyclic.

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

“…Our approach has nothing in common with the ones from [3] or [14]. In [14] it is shown that positive operators commuting with V are not weakly supercyclic and, naturally, the proof employs positivity argument.…”

“…We refer to the survey [18] for the details. Weak supercyclicity was introduced by Sanders [24] and studied in, for instance, [14,15,20,22,25,27]. Gallardo and Montes [8], answering a question raised by Salas, demonstrated that the Volterra operator…”

“…In [20,14] it is shown that V is not weakly supercyclic. In [26] it is proved that for 1 < p < ∞ and any non-zero f ∈ L p [0, 1], the sequence V n f / V n f is weakly convergent to 0 in L p [0,1].…”

“…In [26] it is proved that for 1 < p < ∞ and any non-zero f ∈ L p [0, 1], the sequence V n f / V n f is weakly convergent to 0 in L p [0,1]. In [3] and [14] it is demonstrated that certain operators on L p [0, 1], commuting with V , are not weakly supercyclic. Léon and Piqueras [14] raised a question whether any bounded operator on L p [0, 1] for 1 p < ∞, commuting with V , is not weakly supercyclic.…”

“…In [3] and [14] it is demonstrated that certain operators on L p [0, 1], commuting with V , are not weakly supercyclic. Léon and Piqueras [14] raised a question whether any bounded operator on L p [0, 1] for 1 p < ∞, commuting with V , is not weakly supercyclic. In the present article we close the issue by answering this question affirmatively.…”