Hypercyclic operators on non-locally convex spaces

Abstract

Abstract. We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].During the past years much research has been done about hypercyclic operators; the article [6] contains a rather complete survey of results until 1999. A (continuous linear) operator T : X → X on a topological vector space X is called hypercyclic if it admits a vector x ∈ X having dense orbit Orb(x) = {x, T x, T 2 x, . . .} (x is then called a hypercyclic vector). The following theorem collects some of the recent fundamental results:Theorem. Let X be a locally convex space and let T : X → X be an operator.( A. Peris asked in [8] whether in (3) local convexity is really needed and we now show that it is indeed not:All parts of the Theorem hold for topological vector spaces. The only place in the proof of the Theorem where local convexity plays a role is the following lemma which, for hypercyclic operators, is due to P. Bourdon [3] (the complex case) and J. Bès [2] (the real case). Our proof for the non-locally convex case is quite similar to their arguments.

Common Cesàro hypercyclic vectors

“…Remark 2.8. Observe that Theorem 2.7 (and Theorem 1.2), in view of the result of Wengenroth [44], holds for any topological vector space as well. In particular it holds for the complete metric space H(C) of entire functions, endowed with the topology of uniform convergence on compact sets.…”

“…Observe that Herrero's conjecture implies Ansari's theorem (see for example [38], [14]), while both Herrero's conjecture and Ansari's theorem are corollaries of Bourdon and Feldman's Theorem 2.4 of [14]. Let us also mention that J. Wengenroth [44] was able to relax the hypothesis that X is a Banach space (or more generally a locally convex space). In fact, he showed that the previously mentioned results are valid in arbitrary topological vector spaces.…”

Hypercyclic and mixing operator semigroups

“…The fact that there are no hypercyclic operators on any finite-dimensional topological vector space goes back to Rolewicz [22]. The last result in this direction is due to Wengenroth [26], who proved that a hypercyclic operator on any topological vector space (locally convex or not) has no closed invariant subspaces of positive finite codimension, while any supercyclic operator has no closed invariant subspaces of finite R-codimension greater than 2. In particular, his result implies the (already well known by then) fact that there are no supercyclic operators on a finite-dimensional topological vector space of R-dimension greater than 2.…”

Construction of operators with prescribed orbits in Fréchet spaces with a continuous norm

“…}, for some hypercyclic operator T ∈ L (X) and some hypercyclic vector x ∈ M. We recall the following well-known result: if X is a separable, infinite dimensional Banach space over K, T ∈ L (X) is a hypercyclic operator and x is any hypercyclic vector for T , then K[T ](x) is a dense invariant hypercyclic linear subspace for T , i.e., every non-zero vector of K[T ](x) is hypercyclic for T , see the works of Bourdon [7], Herrero [14], Bès [3] and Wengenroth [19]. Thus, she obtained that every normed space of countable dimension supports an operator which has no non-trivial invariant closed set.…”