Sophie Grivaux

Frequently hypercyclic operators

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.

The rate of convergence in the method of alternating projections

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

A new class of frequently hypercyclic operators

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x in U} has positive lower density. We prove that if T is a bounded operator on X which has sufficiently many eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is automatically frequently hypercyclic.

Sums of hypercyclic operators

Abstract 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.

A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces l(1), c(0) or circle plus(l2) J, and without non-trivial invariant subsets on l(1). We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasi-reflexive dual Banach space which has no non-trivial w*-closed invariant subspace.

Epsilon-hypercyclic operators

Let X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X ; T is hypercyclic if there is a vector x in X with dense orbit under the action ofxa0 T . For a fixed e∈(0,1), we say that T is e-hypercyclic if there exists a vector x in X such that for every non-zero vector y ∈ X there exists an integer n with . The main result of this paper is a construction of a bounded linear operator T on the Banach space l 1 which is e-hypercyclic without being hypercyclic. This answers a question from V.xa0Muller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T.xa0Bermudez, G.xa0Godefroy, K.-G.xa0Grosse-Erdmann and A.xa0Peris. Oberwolfach Rep. 3 (2006), 2227–2276].

Multiples of hypercyclic operators

We give a negative answer to a question of Prajitura by showing that there exists an invertible bilateral weighted shift T on l 2 (Z) such that T and 3T are hypercyclic but 2T is not. Moreover, any G δ set M C (0, oo) which is bounded and bounded away from zero can be realized as M = {t > 0 | tT is hypercyclic} for some invertible operator T acting on a Hilbert space.

Non-recurrence sets for weakly mixing linear dynamical systems

We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space

Invariant subspaces for tridiagonal operators

X

HYPERCYCLIC OPERATORS, MIXING OPERATORS AND THE BOUNDED STEPS PROBLEM

n , which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemanczyk and Rosenblatt, and show in particular that sets