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.     

A note on supercyclic vectors of Hilbert space operators

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists $x\in H$ such that the closure of $\{\alpha T^{n}x:\alpha \in \mathbb{C},n?0\}$ is H, then there is a subsequence ${(n_k)}_{k=1}^{\infty }$ such that the closed linear span of $\{T^{n_k}x:k?1\}$ is not the whole space H.     

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.     

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.     

Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\text{cos}(D)$, $D{e}^D$, or $e^D?aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras.     

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].

     

Topologically transitive skew-products of backward shift operators and hypercyclicity

In this article we look at skew-products of multiples of the backward shift and examine conditions under which the skew-product is topologically transitive or hypercyclic in the second coordinate. We also give an application of the theory to iterated function systems of multiples of backward shift operators.

     

Hypercyclic sequences of PDE-preserving operators

A sequence of continuous linear operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that {T_nx:n0} is dense. A continuous linear operator, acting on some suitable function space, is PDE-preserving for a given set of convolution operators, when it map every kernel set for these operators invariantly. We establish hypercyclic sequences of PDE-preserving operators on , and study closed infinite-dimensional subspaces of, except for zero, hypercyclic vectors for these sequences.     

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.     

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.     

Power bounded operators and supercyclic vectors II

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators.

For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.

     

Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces

By a recent result of M. De La Rosa and C. Read, there exist hypercyclic Banach space operators which do not satisfy the Hypercyclicity Criterion. In the present paper, we prove that such operators can be constructed on a large class of Banach spaces, including or .     

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.     

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.

     

Weakly supercyclic operators

Inspired by a recent result that a weakly hypercyclic operator may fail to be norm hypercyclic, we show there exists a weakly supercyclic operator that fails to be norm supercyclic. Moreover, despite a classical result of Hilden and Wallen that every unilateral weighted backward shift is supercyclic, we show such an operator may have a weakly supercyclic vector that is not a norm supercyclic vector. In addition to these results, we extend a result of Kitai by showing a hyponormal operator cannot be weakly hypercyclic.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.     

Weyl's theorem and hypercyclic/supercyclic operators

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

A note on bounds for non-linear multivalued homogenized operators

In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis.     

Dense invariant sets of common cyclic vectors for Jordan operators

The purpose of this paper is to demonstrate that various collections of cyclic Jordan operators have dense invariant sets of common cyclic vectors.