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...

We characterise the Jacobson radical of an analytic crossed product C 0 (X) × f Z + , answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the Jacobson radical of analytic crossed productsThis consists of all elements whose ''Fourier coefficients'' vanish on the recurrent points of the dynamical system (and the first one is zero). The multidimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.

We show that if a locally compact group G acts properly on a locally compact σ‐compact space X, then there is a family of G‐invariant proper continuous finite‐valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G‐invariant proper compatible metric on X.

Abstract. In this article we answer a question raised by N. Feldman in 2008 concerning the dynamics of tuples of operators on R n . In particular, we prove that for every positive integer n ≥ 2 there exist n-tuples (A 1 , A 2 , . . . , A n ) of n × n matrices over R such that (A 1 , A 2 , . . . , A n ) is hypercyclic. We also establish related results for tuples of 2 × 2 matrices over R or C being in Jordan form.

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on R n is n + 1, thus complementing a recent result due to Feldman.