Hypercyclicity Criterion Research Articles

Frequently hypercyclic random vectors

We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris.

Common frequent hypercyclicity

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are inspired by the classical Frequent Hypercyclicity Criterion and by a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset Λ of the complex plane for the family {λT:λ∈Λ} to have a common frequently hypercyclic vector. In passing, this allows us to exhibit frequently hypercyclic weighted shifts which do not possess common frequently hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common α-frequent hypercyclicity may be vacuous, where the notion of α-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is α-frequently universal for any sequence α satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is α-frequently universal for T, with respect to all such densities α.

Disjoint frequently hypercyclic pseudo-shifts

We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on c0(N) and ℓp(N), 1≤p<∞. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on ℓp(N), this equivalence fails for disjoint versions of these notions.

Disjoint and simultaneously hypercyclic pseudo-shifts

We characterize disjoint and simultaneously hypercyclic tuples of unilateral pseudo-shift operators on ℓp(N). As a consequence, complementing the results of Bernal and Jung, we give a characterization for simultaneously hypercyclic tuples of unilateral weighted shifts. We also give characterizations for unilateral pseudo-shifts that satisfy the Disjoint and Simultaneous Hypercyclicity Criterions. Contrary to the disjoint hypercyclicity case, tuples of weighted shifts turn out to be simultaneously hypercyclic if and only if they satisfy the Simultaneous Hypercyclicity Criterion.

Disjoint and simultaneous hypercyclic Rolewicz-type operators

We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$, which are a generalization of the unilateral backward shift operator. We show that disjoint hypercyclicity and disjoint supercyclicity are equivalent among a subfamily of these operators and disjoint hypercyclic unilateral Rolewicz-type operators always satisfy the Disjoint Hypercyclicity Criterion. We also characterize simultaneous hypercyclic unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$.

Existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing

In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory, 85 (2) (2016), 191-220] by showing the existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing and, therefore, fail to satisfy the Disjoint Hypercyclicity Criterion. We also show that given an operator T such that T⊕T is frequently hypercyclic, the set of operators S such that T,S are disjoint frequently hypercyclic but fail to satisfy the Disjoint Hypercyclicity Criterion is SOT dense in the algebra of bounded linear operators.

The chaotic Black–Scholes equation with time-dependent coefficients

In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line $$[0,\infty )$$ , and we consider the generalized Black–Scholes equation as a linear nonautonomous abstract Cauchy problem. It is shown that this problem is wellposed and an explicit formula for the corresponding strongly continuous evolution family is given. Furthermore, a hypercyclicity criterion for strongly continuous evolution families on separable Banach spaces is proved and applied to the Black–Scholes evolution family. Finally, the chaotic behavior is defined for strongly continuous evolution families and it is shown that the Black–Scholes evolution family is chaotic when the volatility and the interest rates are periodic functions.

Frequent universality criterion and densities

We improve a recent result by giving the optimal conclusion both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of $A$-densities, where $A$ refers to some weighted densities sharper than the natural lower density. Moreover, we construct an operator which is logarithmically frequently hypercyclic but not frequently hypercyclic.

Upper frequent hypercyclicity and related notions

Enhancing a recent result of Bayart and Ruzsa we obtain a Birkhoff-type characterization of upper frequently hypercyclic operators and a corresponding Upper Frequent Hypercyclicity Criterion. As an application we characterize upper frequently hypercyclic weighted backward shifts on sequence spaces, which in turn allows us to come up with various counter-examples in linear dynamics that are substantially simpler than those previously obtained in the literature. More generally, we introduce the notion of upper Furstenberg families $$\mathcal {A}$$ and show that our main results hold for the corresponding $$\mathcal {A}$$ -hypercyclic operators.

Frequently hypercyclic operators with irregularly visiting orbits

We prove that a bounded operator T on a separable Banach space X satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors x∈X such that the set NT(x,U)={n≥1;Tnx∈U} of return times of x into U under the action of T has positive lower density for every non-empty open set U⊆X, but there exists a non-empty open set U0⊆X such that NT(x,U0) has no density.

A quantitative interpretation of the frequent hypercyclicity criterion

We give a quantitative interpretation of the frequent hypercyclicity criterion. Actually we show that an operator which satisfies the frequent hypercyclicity criterion is necessarily $A$-frequently hypercyclic, where $A$ refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the ‘frequency’ measured by the frequent hypercyclicity criterion. Moreover, we construct an example of a unilateral weighted shift which is frequently hypercyclic but not $A$-frequently hypercyclic on a particular scale.

Q-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form $$C_{R,T}(S)=RST$$ to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map $$C_{R}(S)=RSR^*$$ is shown to be q-frequently hypercyclic on the space $$\mathcal {K}(H)$$ of all compact operators and the real topological vector space $$\mathcal {S}(H)$$ of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for $$C_{R,T}$$ to be q-frequently hypercyclic on the Schatten von Neumann classes $$S_p(H)$$ . We also characterize frequent hypercyclicity of $$C_{M^*_\varphi ,M_\psi }$$ on the trace-class of the Hardy space, where the symbol $$M_\varphi $$ denotes the multiplication operator associated to $$\varphi $$ .

Hypercyclicity of the tensor products of multiplication operators on Hardy spaces

In this paper, we give a necessary and sufficient condition for the tensor product of two adjoints of multiplication hypercyclic operators to be mixing and chaotic. These results partially answer a question posed by Félix Martínez-Giménez and Alfredo Peris in 2003. We also prove that the tensor product of an operator satisfying the Frequent Hypercyclicity Criterion with the identity is frequently hypercyclic, thereby strengthening a previous result of Martínez-Giménez and Peris.

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 notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.

A remark on the Frequent Hypercyclicity Criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,\mu)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.

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.

DYNAMICS OF A SPECIAL COMBINATION OF WEIGHTED COMPOSITION OPERATORS ON HILBERT FUNCTION SPACES

In this paper we give some sufficient conditions for the adjoint of a combination of weighted composition operators, acting on some function spaces, satisfying the hypercyclicity criterion. AMS Subject Classification: 47B37, 47B33

Strong mixing measures for $$C_0$$ C 0 -semigroups

Our purpose is to obtain a very effective and general method to prove that certain $$C_0$$ -semigroups admit invariant strongly mixing measures. More precisely, we show that the frequent hypercyclicity criterion for $$C_0$$ -semigroups ensures the existence of invariant strongly mixing measures with full support. We will provide several examples, that range from birth-and-death models to the Black–Scholes equation, which illustrate these results.

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 with 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, T2 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.

TUPLE OF OPERATORS WITH THE PROPERTY OF HYPERCYCLICITY CRITERION

In this paper we give the conditions under which a tuple of oper- ators holds in the Hypercyclicity Criterion.