Some versions of supercyclicity for a set of operators

Abstract

Let X be a complex topological vector space and L(X) the set of all continuous linear operators on X. An operator T ? L(X) is supercyclic if there is x ? X such that, COrb(T,x) = {?Tnx : ? ? C, n ? 0}, is dense in X. In this paper, we extend this notion from a single operator T ? L(X) to a subset of operators ? ? L(X). We prove that most of related proprieties to supercyclicity in the case of a single operator T remains true for subset of operators ?. This leads us to obtain some results for C-regularized groups of operators.

Disjoint strong transitivity of composition operators

“…For more information about hypercyclic, cyclic, and supercyclic operators and their proprieties, see [1,2,3,4,26,7,24].…”

Codiskcyclic sets of operators on complex topological vector spaces

“…For a general overview of the codiskcyclicity, see [12,13,15,18]. Recently, some notions of linear dynamical systems were introduced for a set Γ of operators instead of a single operator T, see [1,2,3,4,5,6]: A set Γ of operators is called hypercyclic if there exists a vector x in X such that its orbit under Γ satisfies Orb(Γ, x) = {T x : T ∈ Γ}, is a dense subset of X. If there exits a vector x ∈ X such that C.Orb(Γ, x) = {αT x : T ∈ Γ, α ∈ C}, is a dense subset of X, then Γ is supercyclic.…”

“…1) we infer thatkz n − α n T t n xk = kz n − α n T t n y n + α n T t n y n − α n T t n xk ≤ kz n − α n T tn y n k + kα n T tn (y n − x)k ≤ kz n − α n T tn y n k + kα n T tn k…”