arXiv · 2206.13075
These notes deal with some recent assertions about truncations $f \mapsto |f|$ and compositions $f \mapsto g\circ f$ in the spaces $A^s_{p,q}(\mathbb{R}^n)$, $A \in \{B,F \}$.
Hans Triebel. 2022-06-27. Truncations and compositions in function spaces. https://arxiv.org/abs/2206.13075
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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$^*$, 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 $(T,T^2,\dotsc,T^k)$ 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.
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