An Extension of Hypercyclicity for N-Linear Operators

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 for N-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclic N-linear operators, for each N≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines support N-linear operators with residual sets of hypercyclic vectors, for N=2.