Typicality of operators on Fréchet algebras admitting a hypercyclic algebra

Abstract

This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra $H(\mathbb{C})$ of entire functions on $\mathbb{C}$, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras $X={\ell }_p(\mathbb{N})$, $1\le p<+\infty$, or $X=c_0(\mathbb{N})$ endowed with the coordinatewise product, and show that whenever $M>1$, a typical operator on $X$ of norm less than or equal to $M$ admits a hypercyclic algebra.

36 pages

References in corpus (1)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples