Abstract
Let X role=presentation> X X X be a complex topological vector space with d i m ( X ) > 1 role=presentation> d i m ( X ) > 1 d i m ( X ) > 1 dim(X)>1 and B ( X ) role=presentation> ( X ) B ( X ) \mathcal{B}(X) the set of all continuous linear operators on X role=presentation> X X X . The concept of hypercyclicity for a subset of B ( X ) role=presentation> ( X ) B ( X ) \mathcal{B}(X) was introduced in [1]. In this work, we introduce the notion of hypercyclic criterion for a subset of B ( X ) role=presentation> ( X ) B ( X ) \mathcal{B}(X) . We extend some results known for a single operator and C 0 role=presentation> C 0 C 0 C_0 -semigroup to a subset of B ( X ) role=presentation> ( X ) B ( X ) \mathcal{B}(X) and we give applications for C role=presentation> C C C -regularized groups of operators.
Highlights
Introduction and preliminaryLet X be a complex topological vector space with dim(X) > 1 and B(X) the set of all continuous linear operators on X
We introduce the notion of quasisimilarity for sets of operators
We prove that hypercyclicity for a set of operators is preserved under quasisimilarity
Summary
Let X be a complex topological vector space with dim(X) > 1 and B(X) the set of all continuous linear operators on X. Bourdon and Feldman [6] proved that if x ∈ HC(T ) , T x ∈ HC(T ) and if p is a nonzero polynomial, p(T ) has dense range These results do not hold for every hypercyclic set of operators. Proposition 2.2 Let Γ ⊂ B(X) be a hypercyclic set and T ∈ B(X) be an operator with dense range. We prove that the same result holds for the set of hypercyclic vectors of a set of operators. Definition 3.1 [1] A set Γ ⊂ B(X) is said to be topologically transitive if for each pair of nonempty open subsets in X there exists some T ∈ Γ such that.
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