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 $N\ge 2$. Indeed, the nonnormable spaces of entire functions and the countable product of lines support $N$-linear operators with residual sets of hypercyclic vectors, for $N=2$.