Hypercyclic Operators Research Articles

Furstenberg theorem for frequently hypercyclic operators

In this paper, we show that if the direct sum T T of frequently hypercyclic operators is frequently hypercyclic, then every higher direct sum T T is also frequently hypercyclic. Mathematics Subject Classication: 47A16

Difference sets and frequently hypercyclic weighted shifts

AbstractWe solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on${\ell }^{p} ( \mathbb{Z} )$,$p\geq 1$. 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$ \mathcal{U} $-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on${c}_{0} $. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

ON POINT SPECTRUM OF SUBSPACE-HYPERCYCLIC OPERATORS

It is well known that if T is a hypercyclic operator, thenp(T � ) = �. We prove in this paper that this is not true for subspace-hypercyclic oper- ators. We show that if T is subspace-hypercyclic, thenp(T � ) may be empty or not. Moreover we show that for every scalarwith |�| > 1, there exists a subspace-hypercyclic operator T such that ||T|| = |�| andp(T � ) 6 �.

Hypercyclic operators on spaces of block-symmetric analytic functions

The paper contains proof of the hypercyclicity of “symmetric translation” on the algebras of block-symmetric analytic functions of bounded type on an isomorphic copy of $l_1$.

Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set Σ(X) of countable dense linearly independent subsets of X. As a consequence, each A∈Σ(X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm.We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on Σ(X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.

Commutation relations and hypercyclic operators

In this paper we establish hypercyclicity of continuous linear operators on \({H(\mathbb{C})}\) that satisfy certain commutation relations.

SYNDETIC SEQUENCES AND DYNAMICS OF OPERATORS

In the present paper, we show that a continuous linear operator T on a Frechet space satisfies the Hypercyclic Criterion with respect to a syndetic sequence must satisfy the Kitai Criterion. On the other hand, an operator, hereditarily hypercyclic with respect to a syndetic sequence must be mixing. We also construct weighted shift operators satisfying the Hypercyclicity Criterion which do not satisfy the Kitai Criterion. In other words, hereditarily hypercyclic operators without being mixing.

Hypercyclic operators on topological vector spaces

Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fr\'echet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.

Weyl's type theorems and hypercyclic operators

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T* ∈▪, then generalized a-Browder's theorem holds for f(T) for every f ∈ Hol(σ(T)). (ii) If T or T* ∈▪ has topological uniform descent at all λ ∈ iso(σ(T)), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈▪ has topological uniform descent at all λ ∈E(T), then T satisfies generalized Weyl's theorem. (iv) Let T ∈▪. If T satisfies the growth condition Gd (d ≥ 1), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈▪, then, f(σSBF+−(T)) = σSBF+−(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈▪. If T − λI has finite ascent at every λ ∈Ea(T) and if F is of finite rank on H such that TF = FT, then T + F obeys generalized a-Weyl's theorem.

Numerically Hypercyclic Operators

An operator T acting on a normed space E is numerically hypercyclic if, for some \({(x, x^*)\in \Pi(E)}\), the numerical orbit\({\{x^*(T^n(x)):n\geq 0\}}\) is dense in \({\mathbb{C}}\). We prove that finite dimensional Banach spaces with dimension at least two support numerically hypercyclic operators. We also characterize the numerically hypercyclic weighted shifts on classical sequence spaces.

Prescribed compressions of dual hypercyclic operators

Prescribed compressions of dual hypercyclic operators

Residuality of Sets of Hypercyclic Operators

We prove a criterion that guarantees that in a given class of operators the set of hypercyclic ones is residual. We also prove the existence of quasinilpotent Volterra composition operators, \({V_\varphi}\) , such that both \({V_\varphi}\) and \({V_\varphi^\star}\) are supercyclic and both \({I + V_\varphi}\) and \({I + V_\varphi^\star}\) are hypercyclic.

Constant-norm scrambled sets for hypercyclic operators

Let T : X → X be a hypercyclic operator of a Banach space X, let D ( T ) = { x ∈ X : x has a dense T -orbit } , and let X r = { x ∈ X : ‖ x ‖ = r } for r > 0 . We show that there is a linearly independent subset S ⊂ D ( T ) with the following properties: (i) for any r > 0 , and any nonempty, relatively open subset U of X r , the intersection S ∩ U is uncountable, (ii) S − S ⊂ D ( T ) ∪ { 0 } ; and in particular, l i m i n f n → ∞ ‖ T n a − T n b ‖ = 0 and l i m s u p n → ∞ ‖ T n a − T n b ‖ = ∞ for any two distinct a , b ∈ S .

Common hypercyclic vectors for the unitary orbit of a hypercyclic operator

For a separable, infinite dimensional Hilbert space, it was recently shown by the authors that the similarity orbit of a hypercyclic operator contains a path of operators which is dense in the operator algebra with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense G δ set. Motivated by that result, we show in the present paper that the unitary orbit of any hypercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given hypercyclic vector is a common hypercyclic vector for the entire path.

A new class of frequently hypercyclic operators

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive lower density. We prove that if T is a bounded operator on X which has sufficiently many eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is automatically frequently hypercyclic.

Dual disjoint hypercyclic operators

Let E be a separable Fréchet space. The operators T 1 , … , T m are disjoint hypercyclic if there exists x ∈ E such that the orbit of ( x , … , x ) under ( T 1 , … , T m ) is dense in E × ⋯ × E . We show that every separable Banach space E admits an m-tuple of bounded linear operators which are disjoint hypercyclic. If, in addition, its dual E ∗ is separable, then they can be constructed such that T 1 ∗ , … , T m ∗ are also disjoint hypercyclic.

Hypercyclic operators on quasi-Mazur spaces

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, Köthe (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.

Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Given a separable, infinite dimensional Hilbert space, it was recently shown by the authors that there is a path of chaotic operators, which is dense in the operator algebra with the strong operator topology, and along which every operator has the exact same dense G δ set of hypercyclic vectors. In the present work, we show that the conjugate set of any hypercyclic operator on a separable, infinite dimensional Banach space always contains a path of operators which is dense with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense G δ set. As a corollary, the hypercyclic operators on such a Banach space form a connected subset of the operator algebra with the strong operator topology.

On the Salas Theorem and Hypercyclicity of f(T)

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 the set of all hypercyclic (mixing, chaotic, frequently hypercyclic, respectively) operators on a Hilbert space coincide. This implies that the set of all hypercyclic operators that do not satisfy the hypercyclicity criterion is rather small—of first category (in the norm-closure of hypercyclic operators).

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 , .