On linear dynamics of sets of operators

Abstract

Let X be a complex topological vector space with dim(X) > 1 and B(X) the set of all continuous linear operators on X. The concept of hypercyclicity for a subset of B(X), was introduced in [1]. In this work, we introduce the notion of hypercyclic criterion for a subset of B(X). We extend some results known for a single operator and C 0 -semigroup to a subset of B(X) and we give applications for C-regularized groups of operators.2000 Mathematics Subject Classification. Primary 47A80, 47A53; secondary 47A10, 47A11. Key words and phrases. Orbit, hypercyclic sets of operators, hypercyclic Operators, C 0 -semigroup, Cregularized group.If T = S, then U ∩ V = ∅ which means that I(U ) ∩ V = ∅. Since I ∈ Γ, the result holds. If T = S, then there exists A ∈ Γ such that T = AS. By (3.1), we have A(Sx) ∈ U and A(Sx) ∈ A(V ), this means that U ∩ A(V ) = ∅. Hence, Γ is a topologically transitive set.Remark 3.8. Assume that X is not necessary without isolated point and let Γ ⊂ B(X) be a hypercyclic set. If Γ satisfies the condition of Theorem 3.7, then Γ ∪ {I} is a topologically transitive set.

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].…”

Some versions of supercyclicity for a set of operators

“…Such a vector x is called a hypercyclic vector for Γ. The set of all hypercyclic vectors for Γ is denoted by HC(Γ), see [2,4]. If the space generated by Orb(Γ, x);…”

Codiskcyclic sets of operators on complex topological vector spaces

“…Recently, some notions of the linear dynamical system 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; Orb(Γ, x) = {T x : T ∈ Γ}, is a dense subset of X. If there exits a vector x ∈ X such that COrb(Γ, x) = {αT x : T ∈ Γ, α ∈ C}, is a dense subset of X, then Γ is supercyclic.…”