Hypercyclic Operators Research Articles

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

Trichotomy for the orbits of a hypercyclic operator on a Banach space

We obtain a trichotomy for the orbits of a hypercyclic operator T T on a separable Banach space X X : (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on ℓ p \ell ^p show that all three cases can happen.

Chaotic and frequently hypercyclic operators in the weighted space of entire functions

Chaotic and frequently hypercyclic operators in the weighted space of entire functions

Extending families of disjoint hypercyclic operators

In the present note, we solve two open questions posed by Salas in [33] about disjoint hypercyclic operators. First, we show that given any family T1,…,TN of disjoint hypercyclic operators, one can always select an operator T such that the extended family T1,…,TN,T of operators remains disjoint hypercyclic. In fact, we prove that the set of operators T which can extend the family of disjoint hypercyclic operators is dense in the strong operator topology in the algebra of bounded operators. Second, we show the existence of two disjoint weakly mixing operators that fail to possess a dense d-hypercyclic manifold. Thus, these operators satisfy the Disjoint Blow-up/Collapse property but fail to satisfy the Strong Disjoint Blow-up/Collapse property, a notion which was introduced by Salas as a sufficient condition for having a dense linear manifold of disjoint hypercyclic vectors.

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

On Hypercyclic Operators in Weighted Spaces of Infinitely Differentiable Functions

Рассматривается инвариантное относительно дифференцирования весовое пространство Фреше ${\mathcal E}(\varphi)$ бесконечно дифференцируемых функций в ${\mathbb R}^n$, порожденное счетным семейством $\varphi$ непрерывных вещественнозначных функций в ${\mathbb R}^n$. При минимальных ограничениях на $\varphi$ показано, что любой линейный непрерывный оператор в пространстве ${\mathcal E}(\varphi)$, коммутирующий с операторами частного дифференцирования и не являющийся скалярным кратным тождественного отображения, является гиперциклическим. Приведены примеры гиперциклических операторов в ${\mathcal E}(\varphi)$ для случаев, когда пространство ${\mathcal E}(\varphi)$ инвариантно относительно сдвига. Библиография: 8 названий.

Multiple recurrence and hypercyclicity

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where $\mathcal F$ is the family of subsets of the natural numbers containing arbitrarily long arithmetic progressions. We prove several properties of hypercyclic multiply recurrent operators, we characterize those operators which are weakly mixing and multiply recurrent, and we show that there are operators that are multiply recurrent and hypercyclic but not weakly mixing.

Property $(W_{E})$ and topological uniform descent

We characterize the property $(W_{E})$ in terms of topological uniform descent, and give the transfer of property $(W_{E})$ from bounded linear operators to functions of operators. Moreover, the perturbation of property $(W_{E})$ under power finite rank operators is investigated. As an application, the property for hypercyclic operators and supercyclic operators is investigated.

Total and non-total suborbits for hypercyclic operators

In this note, it is proved that if X is a separable infinite dimensional Fréchet space that admits a continuous norm then, given a closed infinite dimensional subspace of X, there exists a hypercyclic operator admitting a dense orbit which in turn admits a suborbit all of whose sub-suborbits are total in the prescribed subspace. This is related to a recently published result asserting that every supercyclic vector for an operator on a Hilbert space supports a non-total suborbit. Here we also extend this result to normed spaces.

Disjoint and common hypercyclic algebras

Over time the concept of hypercyclicity has been explored in different manners and contexts, gaining new forms and applications. In particular when the space has an adjacent structure, we can always look for sets of hypercyclic vectors compatible with that framework. In this paper we deal with hypercyclic operators acting on Fréchet sequence algebras and give criteria for the existence of common and disjoint hypercyclic algebras.

On the Dynamics of Lipschitz Operators

By the linearization property of Lipschitz-free spaces, any Lipschitz map $$f : M \rightarrow N$$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $${\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(N)$$ between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps $$f : M \rightarrow M$$ and the linear dynamics of their extensions $${\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(M)$$ . This not only allows us to relate topological dynamical systems to linear dynamical systems but also provide a new class of hypercyclic operators acting on Lipschitz-free spaces.

Ideals of hypercyclic operators that factor through $\ell ^p$

We study the injective and surjective hull of operator ideals generated by hypercyclic backward weighted shifts that factor through $\ell^p$.

Hypercyclic operators for iterated function systems

In this paper we introduce and study the notion of hypercyclicity for iterated function systems (IFS) of operators. We prove that for a linear IFS, hypercyclicity implies sensitivity and if an IFS is abelian, then hypercyclicity also implies multi-sensitivity and hence thick sensitivity. We also give some equivalent conditions for hypercyclicity as well as weakly mixing for an IFS of operators.

Two families of hypercyclic nonconvolution operators

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$, let $C_{\lambda,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{\lambda,b} f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda,b}=C_{\lambda,b} \circ D$ by showing that whenever $|\lambda|\geq 1$, the collection of operators \begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators \begin{align*} \{C_{\lambda,b}\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*} consists entirely of hypercyclic operators.

Frame-cyclic operators and their properties

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

Existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing

In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory, 85 (2) (2016), 191-220] by showing the existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing and, therefore, fail to satisfy the Disjoint Hypercyclicity Criterion. We also show that given an operator T such that T⊕T is frequently hypercyclic, the set of operators S such that T,S are disjoint frequently hypercyclic but fail to satisfy the Disjoint Hypercyclicity Criterion is SOT dense in the algebra of bounded linear operators.

INVERSE OF FREQUENTLY HYPERCYCLIC OPERATORS

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

Supermixing and hypermixing operators

As stronger versions of the mixing property, supermixing and hypermixing notions are introduced and investigated. We show that most of the newly known strongly hypercyclic operators satisfy the stronger condition of being hypermixing. For Banach space operators, we see that some suitable scalar multiples of any non-invertible strongly hypercyclic operator are hypermixing. We prove that on Banach spaces ℓp and c0, a weighted backward shift is supermixing if and only if it is mixing, but, a supermixing weighted backward shift is not necessarily hypermixing.

Chaos and frequent hypercyclicity for weighted shifts

AbstractBayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

Chaotic and hypercyclic operators on solid Banach function spaces

Chaotic and hypercyclic operators on solid Banach function spaces