Recent developments in hypercyclicity

  • K.-G. Grosse-Erdmann, Klaus Floret
  • Published 2003

Abstract

Hypercyclicity is the study of linear operators that possess a dense orbit. Although the first examples of hypercyclic operators date back to the first half of the last century, a systematic study of this concept has only been undertaken since the mid-eighties. Seminal papers like the unpublished but widely disseminated thesis of Kitai [43], a highly original and broad investigation by Godefroy and Shapiro [32] and deep operator-theoretic contributions by Herrero [40], [41] were instrumental in creating a flourishing new area of analysis. The survey [36] of 1999 tried to give a complete synopsis of hypercyclicity and the related area of universality. The intervening years have seen remarkable major advances. In particular, G. Costakis, A. Peris and S. Grivaux have solved two of the five1 problems mentioned in [36]. Additionally, many other noteworthy results have been obtained and a number of foundational issues have been clarified. In this note we want to present some of these new developments. For an updated bibliography on hypercyclicity and related areas such as universal functions, chaotic operators, transitive operators, supercyclic operators or hypercyclic semigroups the interested reader is referred to [37]. A very readable and detailed introduction to hypercyclicity from the point of view of linear dynamics is provided by unpublished notes of J. H. Shapiro [56]. The most general setting for hypercyclicity is that of a (real or complex) topological vector space, which will always be assumed to be Hausdorff. Depending on the result one wants to obtain additional structure

Cite this paper

@inproceedings{GrosseErdmann2003RecentDI, title={Recent developments in hypercyclicity}, author={K.-G. Grosse-Erdmann and Klaus Floret}, year={2003} }