The dynamics of linear operators is a young and rapidly evolving branch of functional analysis. In this book, which focuses on hypercyclicity and supercyclicity, the authors assemble the wide body of theory that has received much attention over the last fifteen years and present it for the first time in book form. Selected topics include various kinds of 'existence theorems', the role of connectedness in hypercyclicity, linear dynamics and ergodic theory, frequently hypercyclic and chaotic operators, hypercyclic subspaces, the angle criterion, universality of the Riemann zeta function, and an introduction to operators without non-trivial invariant subspaces. Many original results are included, along with important simplifications of proofs from the existing research literature, making this an invaluable guide for students of the subject. This book will be useful for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.

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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.

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Abstract. We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on ℓ p (Z), p ≥ 1. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is U-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on c 0 . The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

Abstract. We show that the Bohr radius of the polydisk D n behaves asymptotically as (log n)/n. Our argument is based on a new interpolative approach to the Bohnenblust-Hille inequalities which allows us to prove, among other results, that the polynomial Bohnenblust-Hille inequality is subexponential.

We give conditions for an operator T on a complex separable Banach space X with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non-degenerate invariant Gaussian measure with respect to which it is weak-mixing. The existence of such a measure depends on the geometry of the Banach space and on the possibility of parametrizing the T-eigenvector fields of T in a regular way. We also investigate the connection with frequent hypercyclicity.

Abstract. We prove that the multilinear Bohnenblust-Hille is a particular case of a quite general family of optimal inequalities.