An Extension of Hypercyclicity forN-Linear Operators

Juan Bès,J Alberto Conejero

doi:10.1155/2014/609873

Abstract

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.

Highlights

The study of linear dynamics has attracted the interest of a number of researchers from different areas over the past decades
Despite the several isolated examples in the literature due to Birkhoff [1], MacLane [2], and Rolewicz [3], it was not until the eighties with the unpublished Ph.D. thesis of Kitai [4] and the papers by Beauzamy [5] and by Gethner and Shapiro [6] when the notion of hypercyclicity started to become popular among mathematicians devoted to operator theory and functional analysis
The state of the art on linear dynamics was first described by Grosse-Erdmann in [8] and revisited in [9]; see [10]

Summary

Introduction

The study of linear dynamics has attracted the interest of a number of researchers from different areas over the past decades. We recall that a linear operator T ∈ L(X) is said to be hypercyclic if there exists some x ∈ X such that the orbit Orb(T, x) = {x, Tx, T2x, T3x, . For the case of a bilinear operator, this definition of orbit is simpler than the one used for bihypercyclicity, thanks to the linear order in computing the “iterates” of the initial conditions, as Figure 2 shows. With this new type of orbit, it is natural to consider the following definition.

Existence of Supercyclic N-Linear Operators

Hypercyclic N-Linear Operators on ω

Final Comments