Hypercyclic Subspaces Research Articles

Construction of dense maximal-dimensional hypercyclic subspaces for Rolewicz operators

In this paper, weighted backward shift operators Tw associated to a Schauder basis of a Banach space are considered. These operators are emblematic in the setting of linear chaos in topological vector spaces. In a constructive way, it is shown the existence of a dense linear subspace having maximal dimension, all of whose nonzero members are simultaneously Tw-hypercyclic for every w belonging to a sequence of admissible weights. Our proof does not use any general result about algebraic or topological genericity.

Strongly Mixing Convolution Operators on Fréchet Spaces of Entire Functions of a Given Type and Order

We show that convolution operators on certain spaces of entire functions of a given type and order on Banach spaces are strongly mixing with respect to an invariant Borel probability measure with full support (a stronger property than frequent hypercyclicity). Based on results of S. Muro, D. Pinasco and M. Savransky we also show the existence of frequently hypercyclic entire functions of exponential growth, and the existence of frequently hypercyclic subspaces for such convolution operators.

Existence of common hypercyclic subspaces for the derivative operator and the translation operators

We show that the non-zero multiples of the derivative operator and the non-zero multiples of non-trivial translation operators on the space of entire functions share a common hypercyclic subspace, i.e. a closed infinite-dimensional subspace in which each non-zero vector has a dense orbit for each of these operators.

Q-FREQUENT HYPERCYCLICITY IN AN ALGEBRA OF OPERATORS

We study a notion of q-frequent hypercyclicity of linear maps between the Banach algebras consisting of operators on a separable infinite dimensional Banach space. We derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. Some properties are investigated regarding q-frequently hypercyclic subspaces as shown in [5], [6] and [7]. Finally, we study q-frequent hypercyclicity of tensor products and direct sums of operators.

Non-existence of frequently hypercyclic subspaces for P(D)

We answer a question posed by Bonilla and Grosse-Erdmann by showing that the operators P(D) on the space of entire functions H(C), where D is the differentiation operator and P is a polynomial, do not possess a frequently hypercyclic subspace.

Existence of common and upper frequently hypercyclic subspaces

We provide criteria for the existence of upper frequently hypercyclic subspaces and for common hypercyclic subspaces, which include the following consequences. There exist frequently hypercyclic operators with upper-frequently hypercyclic subspaces and no frequently hypercyclic subspace. On the space of entire functions, each differentiation operator induced by a non-constant polynomial supports an upper frequently hypercyclic subspace, and the family of its non-zero scalar multiples has a common hypercyclic subspace. A question of Costakis and Sambarino on the existence of a common hypercyclic subspace for a certain uncountable family of weighted shift operators is also answered.

Hereditarily hypercyclic subspaces

We say that a sequence of operators $(T_n)$ possesses hereditarily hypercyclic subspaces along a sequence $(n_k)$ if for any subsequence $(m_k)\subset(n_k)$, the sequence $(T_{m_k})$ possesses a hypercyclic subspace. While so far no characterization of the existence of hypercyclic subspaces in the case of Fr\'{e}chet spaces is known, we succeed to obtain a characterization of sequences $(T_n)$ possessing hereditarily hypercyclic subspaces along $(n_k)$, under the assumption that the sequence $(T_n)$ satisfies the Hypercyclicity Criterion along $(n_k)$. We also obtain a characterization of operators possessing a hypercyclic subspace under the assumption that $T$ satisfies the Frequent Hypercyclicity Criterion.

Existence and non-existence of frequently hypercyclic subspaces for weighted shifts

We study the existence and the non-existence of frequently hy- percyclic subspaces in Banach spaces. In particular, we give an example of a weighted shift on lp possessing a frequently hypercyclic subspace and an ex- ample of a frequently hypercyclic weighted shift on lp possessing a hypercyclic subspace but no frequently hypercyclic subspace. The latter example allows us to answer positively Problem 1 posed by Bonilla and Grosse-Erdmann in (Monatsh. Math. 168 (2012)).

Существование гиперциклических подпространств у операторов Теплица

Существование гиперциклических подпространств у операторов Теплица

Hypercyclic subspaces and weighted shifts

We first generalize the results of León-Saavedra and Müller (2006) [10] on hypercyclic subspaces to sequences of operators on Fréchet spaces with a continuous norm. Then we study the particular case of iterates of an operator T and show a simple criterion for having no hypercyclic subspace. Finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces lp or c0, on the space of entire functions and on certain Köthe sequence spaces. We also prove that if P is a non-constant polynomial and D is the differentiation operator on the space of entire functions then P(D) possesses a hypercyclic subspace.

Hypercyclic Subspaces on Fréchet Spaces Without Continuous Norm

Known results about hypercyclic subspaces concern either Fréchet spaces with a continuous norm or the space ω. We fill the gap between these spaces by investigating Fréchet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that \({M \cap {\rm ker} p = \{0\}}\) and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fréchet spaces with a continuous norm and about hypercyclic subspaces on ω. In particular, we show that each infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.

Frequently hypercyclic subspaces

We study the existence of frequently hypercyclic subspaces for a given operator, that is, the existence of closed infinite-dimensional subspaces in which every non-zero vector is frequently hypercyclic. We attack the problem with any of the three methods that have been used for hypercyclic subspaces: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products.

Explicit Constructions of Dense Common Hypercyclic Subspaces

We give an explicit construction of a dense infinite dimensional vector space of hypercyclic vectors for the weighted backward shift Tλ (|λ| > 1). We also develop a technique to construct common hypercyclic vectors for countable families of these operators. The techniques developed here do not rely on the Baire category theorem or any kind of existence proof, as do most approaches to this problem.

Hypercyclic subspaces in Fréchet spaces

In this note, we show that every infinite-dimensional separable Fréchet space admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes the earlier work of several authors.

Hypercyclic subspaces for Fréchet space operators

A continuous linear operator T : X → X is hypercyclic if there is an x ∈ X such that the orbit { T n x } is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H ⊆ X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence ( n i ) and an infinite dimensional closed subspace E ⊆ X such that T is hereditarily hypercyclic for ( n i ) and T n i → 0 pointwise on E. In this note we extend this result to the setting of Fréchet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Fréchet space with a continuous norm admits an operator with a hypercyclic subspace.

Hypercyclic subspaces in omega

We show that any countable family of operators of the form P ( B ) , where P is a non-constant polynomial and B is the backward shift operator on ω, the countably infinite product of lines, has a common hypercyclic subspace.

Common Hypercyclic Subspaces

We give a criterion for a family of operators to have a common hypercyclic subspace. We apply this criterion to a family of homotheties.

Hypercyclic subspaces of a Banach space

Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodriguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodriguez, by first establishing a few elementary results about the algebra of operators on a Banach space.

Supercyclic Subspaces: Spectral Theory and Weighted Shifts

A vector x in a Banach space B is called hypercyclic for a bounded operator T if the orbit {Tnx:n⩾0} is dense in B. If the scalar multiples of the elements in the orbit are dense, then the vector x is called supercyclic. We give a general sufficient condition for a bounded operator on a Banach space to have an infinite-dimensional closed subspace of supercyclic vectors. As a consequence, we also obtain a spectral sufficient condition for the existence of such a subspace for an operator. These results allow us to characterize unilateral and bilateral weighted shifts that have an infinite dimensional closed subspace of supercyclic vectors. Surprisingly, there are weighted shift operators that have supercyclic vectors, but in which all closed subspaces of supercyclic vectors are finite dimensional. Our results complement recent work on hypercyclic subspaces and supercyclic subspaces. They also suggest that the problem of the existence of infinite dimensional closed subspaces of supercyclic vectors is not only different, but also more difficult to handle than the corresponding problem for hypercyclic vectors.

Spectral theory and hypercyclic subspaces

A vector x in a Hilbert space H is called hypercyclic for a bounded operator T : H → H if the orbit {Tnx : n ≥ 1} is dense in H. Our main result states that if T satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for T . The converse is true even if T is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator T to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator T : f(z)→ f(z+1) defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.