Suppose that T is a bounded operator on a nonzero Banach space X . Given a vector x ∈ X , we say that x is hypercyclic for T if the orbit OrbTx = {T x}n is dense in X . Similarly, x is said to be weakly hypercyclic if OrbTx is weakly dense in X . A bounded operator is called hypercyclic or weakly hypercyclic if it has a hypercyclic or, respectively, a weakly hypercyclic vector. It is shown in [...

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on l(Z), p ≥ 1. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is U-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently...

We prove that a continuous linear operator T on a topological vector space X with weak topology is mixing if and only if the dual operator T ′ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space ω due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator T on ω, ...

Let {Tt}t≥0 be a hypercyclic strongly continuous semigroup of operators. Then each Tt (t > 0) is hypercyclic as a single operator, and it shares the set of hypercyclic vectors with the semigroup. This answers in the affirmative a natural question concerning hypercyclic C0-semigroups. The analogous result for frequent hypercyclicity is also obtained.

We consider the transitive linear maps on the operator algebra $B(X)$for a separable Banach space $X$. We show if a bounded linear map is norm transitive on $B(X)$,then it must be hypercyclic with strong operator topology. Also we provide a SOT-transitivelinear map without being hypercyclic in the strong operator topology.

we consider the transitive linear maps on the operator algebra $b(x)$for a separable banach space $x$. we show if a bounded linear map is norm transitive on $b(x)$,then it must be hypercyclic with strong operator topology. also we provide a sot-transitivelinear map without being hypercyclic in the strong operator topology.

In this paper, we introduce subspace-frequently hypercyclic operators. We show that these operators are subspace-hypercyclic and there are subspace-hypercyclic  operators that are not subspace-frequently hypercyclic. There is a criterion like to subspace-hypercyclicity criterion that implies subspace-frequent hypercyclicity and if an operator $T$ satisfies this criterion, then $Toplus T$ is sub...

A continuous operator acting on a topological vector space X is called hypercyclic provided there exists a vector x ∈ X such that its orbit {T nx; n ≥ 1} is dense in X. Such a vector is called a hypercyclic vector for T . The set of hypercyclic vectors will be denoted by HC(T ). The first example of hypercyclic operator was given by Birkhoff, 1929 [3], who shows that the operator of translation...

We study hypercyclicity properties of functions of Banach space operators. Generalizations of the results of Herzog-Schmoeger and Bermudez-Miller are obtained. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro. Furthermore, we show that the norm-closures of t...

We show that there exists an invertible frequently hypercyclic operator on $\ell^1(\mathbb{N})$ whose inverse is not hypercyclic.