arXiv subjects

Valentin Gillet

Publications and source records attributed to Valentin Gillet.

6 recordsLinked to original sources

arXiv2026

Stable invariant measures in linear dynamics

We study the existence of stable invariant measures for operators and strongly continuous semigroups of operators on Banach spaces admitting either a dense bilateral backward orbit or a sufficiently rich family of eigenvectors. These invariant measures are realized as the distributions of stochastic integrals with respect to stable random measures. We also discuss invariant measures with other classes of distributions for such operators and semigroups.

math.DS↗

arXiv2026

Weak limit semigroup in operator theory and ergodic theory

We study the weak limit semigroup of an operator $T$, i.e., the set of all operators being weak limit points of the powers of $T$, in three different but related contexts: Koopman operators of measure-preserving transformations, contractions/isometries/unitaries on separable Hilbert spaces and positive operators on $L^p$-spaces. Hereby we focus on finding large subsets of the weak limit semigroup, in particular in the generic case.

math.FA↗

arXiv2025

Linear dynamics of random products of weighted shifts

The aim of this article is to study the dynamics of random products of weighted shifts on a separable Fr\'echet sequence space. That is, given a measure-preserving dynamical system $(\Omega, \mathcal{F}, \mu, \tau)$, a Fr\'echet sequence space $X$ with a basis $(e_n)_{n \geq 0}$, and a strongly measurable map $T : \Omega \to \mathcal{B}(X)$ taking values in a finite set of weighted shifts on $X$, we study the dynamics of the sequence $(T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega))_{n \geq 1}$ for almost every $\omega \in \Omega$. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every $\omega \in \Omega$, we study some examples on the spaces $X = \ell_p$, $X = c_0$ and $X = H(\mathbb{C})$ involving two shifts, first in the commuting case and then in the non-commuting one.

math.DS↗

arXiv2025

Linear dynamics of random products of operators

We study the linear dynamics of the random sequence $(T_n(.))_{n \geq 1}$ of the operators $T_n(\omega) = T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega), n \geq 1$. These products depend on an ergodic measure-preserving transformation $\tau : \mathbb{T} \to \mathbb{T}$ on the probability space $(\mathbb{T}, m)$ and on a strongly measurable map $T : \mathbb{T} \to \mathcal{B}(X)$, where $X$ is a separable Fr\'echet space. We will be focusing on the case where $T(\omega)$ is equal to an operator $T_1$ on $X$ for every $\omega \in A_1$ and equal to an operator $T_2$ on $X$ for every $\omega \in A_2$, where $A_1, A_2$ are two disjoint Borel subsets of $[0,1)$ such that $A_1 \cup A_2 = [0,1)$ and $m(A_k) > 0$ for $k = 1,2$. More precisely, we will be focusing on the case where the operators $T_1$ and $T_2$ are adjoints of multiplication operators on the Hardy space $H^2(\mathbb{D})$, as well as the case where $T_1$ and $T_2$ are entire functions of exponential type of the derivation operator on the space of entire functions. Finally, we will study the linear dynamics of a case of a random product $T_n(\omega)$ for which the operators $T(\tau^i \omega), i \geq 0$, do not commute. We will give particular importance to the case where the ergodic transformation is an irrational rotation or the doubling map on $\mathbb{T}$.

math.FA↗

arXiv2024

Similar operator topologies on the space of positive contractions

In this article, we study the similarity of the Polish operator topologies $\texttt{WOT}$, $\texttt{SOT}$, $\texttt{SOT}\mbox{$_{*}$}$ and $\texttt{SOT}\mbox{$^{*}$}$ on the set of the positive contractions on $\ell_p$ with $p > 1$. Using the notion of norming vector for a positive operator, we prove that these topologies are similar on $\mathcal{P}_1(\ell_2)$, that is, they have the same dense sets in $\mathcal{P}_1(\ell_2)$. In particular, these topologies will share the same comeager sets in $\mathcal{P}_1(\ell_2)$. We then apply these results to the study of typical properties of positive contractions on $\ell_p$-spaces in the Baire category sense. In particular, we prove that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT})$ has no eigenvalue. This stands in strong contrast to a result of Eisner and M\'atrai, stating that the point spectrum of a typical contraction $T \in (\mathcal{B}_1(\ell_2), \texttt{SOT})$ contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{WOT})$ (resp. $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT}\mbox{$_{*}$})$) has no eigenvalue.

math.FA↗

arXiv2024

Typical properties of positive contractions and the invariant subspace problem

In this paper, we first study some elementary properties of a typical positive contraction on $\ell_q$ for the Strong Operator Topology and the Strong* Operator Topology. Using these properties, we prove that a typical positive contraction on $\ell_1$ (resp. on $\ell_2$) has a non-trivial invariant subspace for the Strong Operator Topology (resp. for the Strong Operator Topology and the Strong* Operator Topology). We then focus on the case where $X$ is a Banach space with a basis. We prove that a typical positive contraction on a Banach space with an unconditional basis has no non-trivial closed invariant ideals for the Strong Operator Topology and the Strong* Operator Topology. In particular, this shows that when $X = \ell_q$ with $1 \leq q < \infty$, a typical positive contraction $T$ on $X$ for the Strong Operator Topology (resp. for the Strong* Operator Topology when $1 < q < \infty$) does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion, that is, there is no non-zero positive operator in the commutant of $T$ which is quasinilpotent at a non-zero positive vector of $X$. Finally, we prove that, for the Strong* Operator Topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion.

math.FA↗