A short proof of existence of disjoint hypercyclic operators

We give a short proof of existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Fréchet space. Similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.     

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R₊×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H^∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H²(D)→H²(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable.     

Frequently hypercyclic operators

We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators on separable complex -spaces: is frequently hypercyclic if there exists a vector such that for every nonempty open subset of , the set of integers such that belongs to has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.

     

Porosity and hypercyclic operators

We study if the set of hypercyclic vectors of a hypercyclic operator is the complement of a -porous set. This leads to interesting results for both points of view: a limitation of the size of hypercyclic vectors, and new examples of first category sets which are not -porous.

     

Remarks on a result about hypercyclic non-convolution operators

In this paper we correct a proof by Aron and Markose in [R. Aron, D. Markose, On universal functions, J. Korean Math. Soc. 41 (2004) 65-76] for the hypercyclicity of the operator given by , in the case b≠0.     

Topologically mixing hypercyclic operators

Let be a separable Fréchet space. We prove that a linear operator satisfying a special case of the Hypercyclicity Criterion is topologically mixing, i.e. for any given open sets there exists a positive integer such that for any We also characterize those weighted backward shift operators that are topologically mixing.

     

Disjoint hypercyclic linear fractional composition operators

We characterize disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc, answering a question of Bernal-González. We also study mixing and disjoint mixing behavior of projective limits of endomorphisms of a projective spectrum. In particular, we show that a linear fractional composition operator is mixing on the projective limit of the Sv spaces strictly containing the Dirichlet space if and only if the operator is mixing on the Hardy space.     

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators _T1,_T2,…,_TN$T_1,T_2,\dots ,T_N$ with N?2$N?2$ which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators _T1$T_1$, _T2$T_2$ whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

Dual subspaces of operators

In this note we introduce some new constructions of dual spaces of operators, which are, of course, at the same time, operator spaces in the sense of Pisier (2003) [12]. We exemplify the utility of these constructs by establishing, in this more general setting, a curious and little known result from the theory of dual algebras, namely from Chevreau and Pearcy (1991) [11].     

Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator is hypercyclic if there is an such that the orbit is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces does contain such an operator. We prove that for any infinite-dimensional Banach space with a shrinking symmetric basis, such as and any , there is an operator , where both and are hypercyclic.

     

Disjointness in hypercyclicity

We introduce a notion of disjointness for finitely many hypercyclic operators acting on a common space, notion that is weaker than Furstenberg's disjointness of fluid flows. We provide a criterion to construct disjoint hypercyclic operators, that generalizes some well-known connections between the Hypercyclicity Criterion, hereditary hypercyclicity and topological mixing to the setting of disjointness in hypercyclicity. We provide examples of disjoint hypercyclic operators for powers of weighted shifts on a Hilbert space and for differentiation operators on the space of entire functions on the complex plane.     

Tensor products of recurrent hypercyclic semigroups

We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.     

Hereditarily hypercyclic operators and mixing

We show that hereditary transitivity (respectively strongly hereditary transitivity) is equivalent to weak mixing (respectively strong mixing) in a discrete dynamical system with Polish phase space. We also study the connection between local orbit structure and hypercyclicity, and obtain a “local hypercyclicity criterion.”     

Topological mixing and Hypercyclicity Criterion for sequences of operators

For a sequence of continuous linear operators on a separable Fréchet space , we discuss necessary conditions and sufficient conditions for to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect to the full sequence implies topological mixing; 3) topological mixing implies the Hypercyclicity Criterion with respect to some sequence that cannot be syndetic in general, and also implies condition (b) of the Hypercyclicity Criterion with respect to the full sequence. Applications to two examples of operators on the Fréchet space of entire functions are also discussed.

     

Interpolation by hypercyclic functions for differential operators

We prove that, given a sequence of points in a complex domain Ω without accumulation points, there are functions having prescribed values at the points of the sequence and, simultaneously, having dense orbit in the space of holomorphic functions on Ω. The orbit is taken with respect to any fixed nonscalar differential operator generated by an entire function of subexponential type, thereby extending a recent result about MacLane-hypercyclicity due to Costakis, Vlachou and Niess.     

Weyl type theorem and hypercyclic operators

Using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for T for which the a-Browder's theorem or the a-Weyl's theorem holds. Also, the relation between hypercyclic operators (or supercyclic operators) and the operators which satisfy Weyl type theorem is discussed.     

On supercyclicity of operators from a supercyclic semigroup

We show that for every supercyclic strongly continuous operator semigroup {Tt}_t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup.     

Complete systems of partial derivatives of entire functions and frequently hypercyclic operators

We find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space H(C^d)$H({\mathbb{C}}^d)$ of all entire functions of d   variables. As an application of this result we describe new classes of frequently hypercyclic operators on H(C^d)$H({\mathbb{C}}^d)$.     

Hypercyclic operators on non-locally convex spaces

We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].