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 $X=\ell_p$, $1&lt; p&lt;\infty$. We show that, with respect to SOT$<sup>*$, 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 $(T,T<sup>2,\dotsc,T<sup>k)$ is disjoint hypercyclic for any $k\geq 2$. We also study the typical dynamical properties for the concrete family $$\mathcal{M}={I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})}$$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.