arXiv subjects
Publications and source records attributed to Frédéric Bayart.
This paper is devoted to an in-depth study of the minimal commutant property for composition operators acting on Hilbert spaces of holomorphic functions in several complex variables, such as the Fock space on $\mathbb{C}^d$, the Hardy space on the Euclidean ball, and the Hardy space on the unit polydisc.
We study the dynamical properties of composition operators acting on Banach spaces of measurable functions. In particular, we study in some detail the composition operators induced by odometers, which allows us to give a variety of new examples and counter-examples. We also get general statements about supercyclicity and frequent hypercyclicity of composition operators on $L\_p$-spaces.
We study composition operators between weighted Bergman spaces of the polydisc induced by smooth symbols. We prove a general result of continuity which only involves the behaviour of the symbol on the polytorus. We deduce from this several consequences about the automatic continuity of the induced operator. We study in depth the case of the tridisc and exhibit several examples showing that a characterization of continuity using only derivatives seems impossible.
In this study we consider the approximation numbers of differences of composition operators acting on the Hardy-Hilbert space H 2 (D). We obtain both upper and lower bounds for these approximation numbers and by applying these general results to composition operators with specific types of symbols, we demonstrate the effect of boundary behaviour over the approximation numbers. Moreover, we use these one-dimensional methods and examples to understand the approximation numbers of differences of composition operators acting on the space H 2 (D 2 ) of the bidisc.
Let $φ$ be a holomorphic map which is a symbol of a bounded composition operator $C_φ$ acting on the Hardy-Hilbert space of Dirichlet series. We find a Königs map for $φ$. We then deduce several applications on $C_φ$ (e.g. on its spectrum, on its dynamical properties). In particular, we study for a large class of symbols $φ$ if the associated composition operator has a minimal commutant.
In the present work, we establish sufficient conditions for a Dirichlet series induced by general frequencies to be universal with respect to vertical translations. Our results can be applied to known universal objects such as Hurwitz zeta functions and also can provide new examples of universal Dirichlet series including the alternating prime zeta function $\sum_{n\geq1}(-1)^np_n^{-s}$.
We give a necessary and sufficient condition for a holomorphic self-map $ϕ$ of the tridisc to induce a bounded composition operator on the associated Hardy space. This condition depends on the behaviour of the first and the second derivative of the symbol at boundary points. We also discuss compactness of composition operators on the bidisc and the tridisc.
We compute the essential norm of inclusion operators, composition operators and multipliers acting from a closed subspace of some $L^p$-space into a subspace of some $L^q$-space, with $p > q.$
We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardy space $\mathcal{H}^2$ of Dirichlet series. For $p\geq 2$, these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for $\mathcal{H}^2$ and we give examples of bounded composition operators with Dirichlet series symbols on $\mathcal{H}^p,\,p>0$.
For every fixed $ε$ $\in$ (0, 1), we construct an operator on the separable Hilbert space which is $δ$-hypercyclic for all $δ$ $\in$ ($ε$, 1) and which is not $δ$-hypercyclic for all $δ$ $\in$ (0, $ε$).
We give a sufficient condition for two operators to be disjointly frequently hypercyclic. We apply this criterion to composition operators acting on $H(\mathbb D)$ or on the Hardy space $H^2(\mathbb D)$. We simplify a result on disjoint frequent hypercyclicity of pseudo shifts of a recent paper of Martin et al. and we exhibit two disjointly frequently hypercyclic weighted shifts.
In this paper we provide a full characterization of cyclic composition operators defined on the d-dimensional Fock space $\mathcal F(\mathbb C^d)$ in terms of their symbol. Also, we study the supercyclicity and convex-cyclicity of this type of operators. We end this work by computing the approximation numbers of compact composition operators defined on $\mathcal F(\mathbb C^d)$.
We give sufficient conditions for a general Dirichlet series to be universal with respect to translations or rearrangements.
We give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series.
The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.
We investigate the existence of a common hypercyclic vector for a family $(T_λ)_{λ\in Λ}$ of hypercyclicoperators acting on the same Banach space $X$. We give positive and negative results involving the dimension of $Λ$ and the regularity of each map $λ\in Λ\mapsto T_λ^n x$, $x\in X$, $n\in\mathbb N$.
We show that an invertible bilateral weighted shift is strongly structurally stable if and only if it has the shadowing property. We also exhibit a K{ö}the sequence space supporting a frequently hypercyclic weighted shift, but no chaotic weighted shifts.
Given a frequency $λ$, we study general Dirichlet series $\sum a_n e^{-λ_n s}$. First, we give a new condition on $λ$ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to get general results on these spaces as spaces of holomorphic functions.