Typical dynamical properties of operators on ${\ell }_p$

Abstract

We investigate the typical dynamical properties of hypercyclic operators in ${\mathcal{L}}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X={\ell }_p$, $1<p<\infty$. We show that, with respect to SOT${}^{\ast }$, a typical operator $T\in {\mathcal{L}}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dots ,T^k)$ is disjoint hypercyclic for any $k\ge 2$. We also study the typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X):w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

29 pages