Invariant Gaussian measures for operators on Banach spaces and linear dynamics

Bayart, Frédéric; Grivaux, Sophie

doi:10.1112/plms/pdl013

Cited by 83 publications

(134 citation statements)

References 26 publications

Supporting

Mentioning

133

Contrasting

Unclassified

Order By: Relevance

“…In fact, in this case, T is frequently hypercyclic; see [6,8,31,32]. The point of Lemma 2.17 is that if we only assume the weaker hypothesis (2.18) we can still conclude that T is recurrent.…”

Section: Remark 219 a Few Remarks Are In Ordermentioning

confidence: 85%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

Abstract. We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. The most studied notion in linear dynamics is that of hypercyclicity: a bounded linear operator T acting on a separable Banach space is hypercyclic if there exists a vector whose orbit under T is dense in the space. On the other hand, a very central notion in topological dynamics is that of recurrence. This notion goes back to Poincaré and Birkhoff and it refers to the existence of points in the space for which parts of their orbits under a continuous map "return" to themselves. The purpose of this note is the study of the notion of recurrence, together with its variations, in the context of linear dynamics. Some examples and characterizations of recurrence for special In an effort to characterize recurrent linear operators one many times falls back to the notion of hypercyclicity. This is for example the case when we study the recurrence properties of backwards shifts, say on ℓ 2 (Z). The reason behind is that, according to a result of Seceleanu, [51], the orbits of these operators satisfy a zero-one law: if the orbit of a weighted backward shift contains a non-zero limit point then the corresponding shift is actually hypercyclic. Thus a weighted backward shift on ℓ 2 (Z) is recurrent if and only if it is hypercyclic. The same equivalence is true, albeit for different reasons, for the adjoint of a multiplication operators on the Hardy space H 2 (D). These connections to hypercyclicity, already observed in [17], come up naturally and thus motivate a further search on whether the properties of recurrent operators resemble the properties of hypercyclic ones, in general. It turns out that, indeed, there are many structural similarities between the set of hypercyclic vectors and the set of recurrent vectors in the sense that they exhibit the same invariances. Furthermore, the spectral properties of hypercyclic and recurrent operators are somewhat similar, although this vague statement should be interpreted with some care. However, these similarities cannot be pushed too much as there are obviously many classes of operators which are recurrent without being hypercyclic. One can find such examples among composition operators on the Hardy space H 2 (D). However, the primordial example is given just by considering unimodular multiples of the identity operator. A more general class for which one needs to address the recurrence properties independently of hypercyclicity is that of unitary operators on Hilbert spaces.The discussion above hopefully justifies why we will shortly recall a full set of definitions rel...

show abstract

Section: Remark 219 a Few Remarks Are In Ordermentioning

confidence: 85%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…Since several points in the argument in [21] are rather difficult to follow, we give below a complete self-contained exposition of a proof of Theorem 3.22. This new proof also has the advantage that it can be extended to the Banach space setting under suitable assumptions on the geometry of the space; see [6]. As for Proposition 3.18, a proof is sketched in the announcement [20], and Section 3.3 below, which is mostly expository, expands on this.…”

Section: Frequently Hypercyclic Operators 5099mentioning

confidence: 99%

Frequently hypercyclic operators

Bayart

Grivaux

2006

Trans. Amer. Math. Soc.

Self Cite

211

224

View full text Add to dashboard Cite

Abstract. We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators T on separable complex F -spaces: T is frequently hypercyclic if there exists a vector x such that for every nonempty open subset U of X, the set of integers n such that T n x belongs to U 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.

show abstract

“…For example, in [3] the authors constructed a frequently hypercyclic operator on c 0 that is not Devaney chaotic (and not mixing). Is this operator distributionally chaotic?…”

Section: Miscellaneamentioning

confidence: 99%

Distributional chaos for linear operators

Bernardes

Bonilla

Müller

et al. 2013

Journal of Functional Analysis

135

View full text Add to dashboard Cite

ElsevierPeris Manguillot, A.; Bernardes, NC.; Müller, V. (2013) AbstractWe characterize distributional chaos for linear operators on Fréchet spaces in terms of a computable condition (DCC), and also as the existence of distributionally irregular vectors. A sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented, which is very general and can be applied to many classes of operators. Distributional chaos is also analyzed in connection with frequent hypercyclicity, and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results.

show abstract

Invariant Gaussian measures for operators on Banach spaces and linear dynamics

Cited by 83 publications

References 26 publications

Recurrent Linear Operators

Recurrent Linear Operators

Frequently hypercyclic operators

Distributional chaos for linear operators

Contact Info

Product

Resources

About