arXiv subjects
Publications and source records attributed to João V. A. Pinto.
For a fixed $0<\varepsilon<1$, F. Bayart asked in 2024 whether there exists an operator $T$ such that, for every $0<δ<1$, $T$ is $δ$-hypercyclic if and only if $δ\in[\varepsilon,1)$. We answer this question affirmatively by constructing a weighted backward shift on $\ell_2(\mathbb N_0,\ell_2(\mathbb N_0))$ with this property.
In this work, we characterize the shadowing property for weighted shifts on a general class of Banach sequence spaces, extending a theorem proved by Bernardes et al. in ETDS (2020). We then apply this result to characterize this property for weighted composition operators acting on a class of Banach spaces of continuous functions. As an application, we present a characterization of shadowing for bilateral weighted translations on the spaces $C_b(\mathbb{R})$ and $C_0(\mathbb{R})$. In a more general setting, we provide sufficient conditions for weighted composition operators to have the shadowing property. These conditions allow us to obtain complete characterizations of this property for multiplication operators on the Hardy spaces $H^p(\mathbb{D})$, $p\in[1,\infty]$, as well as on spaces of the form $H^\infty(Ω)$ or $C_b(Θ)$.
We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space $X$, endowed with the sup norm, and the locally convex spaces $C(X)_c$ of continuous scalar-valued functions on a completely regular space $X$, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(μ)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(μ)$ spaces and apply it to the study of several dynamical properties.
In this paper, we give characterizations of distributional chaos and mean Li$-$Yorke chaos for weighted backward shifts acting on general Fréchet sequence spaces. As an application, we derive criteria for these two types of chaos in the setting of Köthe sequence spaces $λ_p(A,J)$ for $p \in \{0\}\cup [1, \infty)$ and $J=\mathbb{N}$ or $J=\mathbb{Z}$.
We study the properties of power-boundedness, Li-Yorke chaos, distributional chaos, absolutely Cesàro boundedness and mean Li-Yorke chaos for weighted composition operators on $L^p(μ)$ spaces and on $C_0(Ω)$ spaces. We illustrate the general results by presenting several applications to weighted shifts on the classical sequence spaces $c_0(\mathbb{N})$, $c_0(\mathbb{Z})$, $\ell^p(\mathbb{N})$ and $\ell^p(\mathbb{Z})$ ($1 \leq p < \infty$) and to weighted translation operators on the classical function spaces $C_0[1,\infty)$, $C_0(\mathbb{R})$, $L^p[1,\infty)$ and $L^p(\mathbb{R})$ ($1 \leq p < \infty$).