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.

     

Pathological hypercyclic operators

We exhibit a hypercyclic operator whose square is not hypercyclic. Our operator is necessarily unbounded since a result of S. Ansari asserts that powers of a hypercyclic bounded operator are also hypercyclic. We also exhibit an unbounded Hilbert space operator whose non-zero vectors are hypercyclic. Received: 19 March 2005; revised: 18 July 2005     

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.

     

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.

     

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.

     

Sums of hypercyclic operators

Every bounded operator on a complex infinite-dimensional separable Hilbert space can be written as the sum of two hypercyclic operators, and also as the sum of two chaotic operators.     

Dual disjoint hypercyclic operators

Let E be a separable Fréchet space. The operators T₁,…,Tm are disjoint hypercyclic if there exists x∈E such that the orbit of (x,…,x) under (T₁,…,Tm) 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 are also disjoint hypercyclic.     

Recurrence properties of hypercyclic operators

We generalize the notions of hypercyclic operators, \(\mathfrak {U}\)-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely \(\mathcal {A}\)-hypercyclicity. We then state an \(\mathcal {A}\)-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the \(\mathcal {A}\)-hypercyclicity for weighted shifts. We also investigate which density properties can the sets \({N(x, U)=\{n\in \mathbb {N}\ ; \ T^nx\in U\}}\) have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.     

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.”     

Commutation relations and hypercyclic operators

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

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.     

On hypercyclic operators on Banach spaces

We provide in this paper a direct and constructive proof of the following fact: for a Banach space there are bounded linear operators having hypercyclic vectors if and only if is separable and dim. This is a special case of a recent result, which in turn solves a problem proposed by S. Rolewicz.

     

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.     

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.     

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's theorem and hypercyclic/supercyclic operators

Necessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy are proved.     

Common hypercyclic vectors for families of operators

We provide a criterion for the existence of a residual set of common hypercyclic vectors for an uncountable family of hypercyclic operators which is based on a previous one given by Costakis and Sambarino. As an application, we get common hypercyclic vectors for a particular family of hypercyclic scalar multiples of the adjoint of a multiplier in the Hardy space, generalizing recent results by Abakumov and Gordon and also Bayart. The criterion is applied to other specific families of operators.

     

Weyl type theorems and hypercyclic operators II

In this note, the relation between hypercyclic operator matrices (or supercyclic operator matrices) and the operator matrices which satisfy Weyl type theorems is discussed. Also, using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for for which a-Browder's theorem or a-Weyl's theorem holds.