Typical dynamical properties beyond classical sequence spaces

Determine for which Banach spaces the typicality results for weak mixing, failure of topological ergodicity, weak disjointness, and disjoint hypercyclicity established for operators on \(\ell_p\), \(1<p<\infty\), also hold.

Background

The paper proves that, in the strong-star operator topology on bounded operator balls over $\ell_p$, typical operators are weakly mixing, are not topologically ergodic, are weakly disjoint from a fixed hypercyclic operator, and have strong disjoint-hypercyclicity properties involving their powers. Analogous extensions are briefly noted for $c_0$ and $\ell_1$ under appropriate operator topologies.

The authors explicitly ask whether these typicality phenomena extend to other Banach spaces. The problem seeks to identify the class of Banach spaces for which the same Baire-category dynamical behavior can be established.