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Maëva Ostermann

Publications and source records attributed to Maëva Ostermann.

9 recordsLinked to original sources
arXiv2026

Schauder Basis with Finite Blaschke Products

We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains valid when $Hol(\mathbb D)$ is equipped with a broader class of norms satisfying natural structural conditions. These conditions are satisfied by norms of classical function spaces such as the Hardy spaces $H^p$ ($1\leq p\leq \infty$), the weighted Bergman spaces $A_α^p$ ($1\leq p\leq \infty$, $α>-1$), and BMOA. We also establish the optimality of this framework by proving that such a basis cannot exist in larger spaces, such as the Hardy space $H^p$ and the disc algebra $A(\mathbb D)$.

math.CV↗
arXiv2026

Hypercyclicity of Toeplitz operators with smooth symbols

This paper is devoted to the study of the dynamics of Toeplitz operators $T_F$ with smooth symbols $F$ on the Hardy spaces of the unit disk $H^p$, $p>1$. Building on a model theory for Toeplitz operators on $H^2$ developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for $T_F$ to be hypercyclic on $H^p$. In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators.

math.FA↗
arXiv2026

Embedding of Toeplitz operators with smooth symbols into strongly continuous semigroups

Using the model theory for Toeplitz operators with smooth symbols developed by the fourth author in the 80's, we study whether such operators $T_{F}$ can be embedded into a $C_{0}$-semigroup of operators on the Hardy space $H^p$ of the open unit disk, $1<p<\infty$. We show that it is the case as soon as $0$ belongs to the unbounded connected component of $\mathbb{C}$ minus the interior of the spectrum of $T_{F}$. We provide several conditions on the symbol $F$, both geometric and analytic in nature, ensuring that this sufficient condition is also necessary. For a certain class of symbols, where the curve $F(\mathbb{T})$ is a ``figure eight in a loop" such that $\mathbb{C}\setminusσ(T_F)$ has a bounded connected component, we obtain a complete characterization of the embeddability of $T_F$ into a $C_0$-semigroup. In the last part of the paper, we discuss the embeddability of $T_F$ when the symbol $F$ is not necessarily smooth, using connections with the numerical range and the functional calculus for bounded sectorial operators.

math.FA↗
arXiv2025

Weighted composition operators on de Branges-Rovnyak spaces

In this paper, we characterize the boundedness and the compactness of weighted composition operators acting on a de Branges-Rovnyak space $\mathcal H(b)$, where the symbol $b$ is a rational function in the unit ball of $H^\infty$ that is not a finite Blaschke product. Our results extend those of [2] by exploiting a close relationship between weighted composition operators on $\mathcal H(b)$ and their counterparts on the Hardy space $H^2$.

math.CV↗
arXiv2022

On the image of the mean transform

Let $B(H)$ be the algebra of all bounded operators on a Hilbert space $H$. Let $T=V|T|$ be the polar decomposition of an operator $T\in B(H)$. The mean transform of $T$ is defined by $M(T)=\frac{T+|T|V}{2}$. In this paper, we discuss several properties related to the spectrum, the kernel, the image, the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators as positive, normal, unitary, hyponormal and co-hyponormal operators.

math.FA↗
arXiv2022

The Aluthge and the mean transforms of $m$-isometries

Let $T\in B(H)$ be a bounded linear operator on a Hilbert space $H$, let $T = V|T|$ be its polar decomposition of $T$ and let $λ\in [0,1]$. The $λ$-Aluthge transform $Δ_λ(T)$ and the mean transforms $M(T)$ are defined respectively by: \[Δ_λ(T):=|T|^λV|T|^{1-λ} \;\; \text{and} \;\; M(T):=\frac12(|T|V+V|T|).\] In this paper, we use several examples of weighted shift operators to prove that the Aluthge and mean transforms do not preserve the class of $m-$isometries in any directions.

math.FA↗
arXiv2022

An abstract approach to the Crouzeix conjecture

Let $A$ be a uniform algebra, $θ:A\to M_n(\mathbb{C})$ be a continuous homomorphism and $α:A\to A$ be an antilinear contraction such that \[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that $\|θ\|\le 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that, if further $α(1)=1$, then we may conclude that $\|θ\|\le2$. This would yield a positive solution to the Crouzeix conjecture on numerical ranges. In support of our conjecture, we prove that it is true in two special cases. We also discuss a completely bounded version of our conjecture that brings into play ideas from dilation theory.

math.FA↗
arXiv2021

Pseudospectra and Simultaneous Power Control

We prove that, for $M>0$ and $n,m\ge2$, we can find two $10\times10$ matrices $A$ and $B$ with identical pseudospectra such that we have simultaneously $\|A^n\|/\|B^n\|>M$ and $\|A^m\|/\|B^m\|>M$. We also prove that, under certain conditions, this result holds for two more general functions of $A$ and $B$.

math.FA↗