arXiv subjects

Clément Coine

Publications and source records attributed to Clément Coine.

18 recordsLinked to original sources

arXiv2026

On absolutely Cesàro bounded operators

We study $p$-absolutely Cesàro bounded operators, with particular emphasis on self-improvement, weighted shifts, and linear dynamics. Our first main result shows that, for $1<p<\infty$, every positive absolutely Cesàro bounded operator on $L^p(Ω)$ is automatically $p$-absolutely Cesàro bounded. We then characterize $q$-absolute Cesàro boundedness of backward shifts on weighted $\ell^p$-spaces for $q\geq p$ and use this characterization to construct examples exhibiting a wide range of possible growth rates of the powers. In the dynamical direction, we obtain new obstructions to absolute and strong Cesàro boundedness; in particular, no strongly Cesàro bounded operator on a nonzero Banach space is chaotic or upper frequently hypercyclic. We also introduce Cesàro ratio-boundedness and compare it with absolute Cesàro boundedness. Finally, a general Baire-category principle yields a genericity theorem for the failure of $p$-absolute Cesàro boundedness.

math.FA↗

arXiv2026

Schur tests and estimates for positive semigroups on non commutative $L_p$

We extend the recent results of Arnold on estimates for Kreiss positive semigroups on $L_p$-spaces to the non commutative setting, $1<p<\infty$. We also provide a sharp example. All the results are achieved thanks to the use of the Schur test, including a new version for von Neumann algebras.

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arXiv2026

Invariant subspaces for free linearizations of Lipschitz maps

We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations $\widehat f$ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators $T_{f,e}$ which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every $\widehat f$ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for $T_{f,e}$ under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.

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arXiv2026

Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas

In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class $\mathcal{S}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten $\mathcal{S}^p$-perturbation, $1<p<\infty$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case.

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arXiv2026

Noncommutative $L_p$-differentiability and trace formulae

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace $τ$, and let $L_p(\mathcal{M})$ denote the associated noncommutative $L_p$-space for $1<p<\infty$. Let $n\in\mathbb{N}$ and let $a, b$ be $τ$-measurable self-adjoint operators such that $b\in L_p(\mathcal{M})\cap L_{np}(\mathcal{M})$. For a function $f\in C^n(\mathbb{R})$ whose derivatives $f^{(k)}$ are bounded for $1\le k\le n$, we prove that the map $ϕ:t\in\mathbb{R}\mapsto f(a+tb)-f(a)$ is $n$-times differentiable in the $\|\cdot\|_{L_p}$-norm. This strengthens the corresponding result of de Pagter and Sukochev for $p\neq 2$ and extends it to higher-order derivatives. In addition, if $f^{(n)}\in C_0(\mathbb{R})$ or $b\in \mathcal{M}$, then $ϕ^{(n)}$ is continuous on $\mathbb{R}$. Consequently, we extend the Potapov--Skripka--Sukochev higher-order trace formula from bounded $L_n$-perturbations to not necessarily bounded perturbations in $L_n(\mathcal{M})\cap L_{n^{2}}(\mathcal{M})$. Moreover, we show that this trace formula holds for a broader class of admissible functions than the classes previously considered in the literature.

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arXiv2024

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space $\hilh$, denoted as $\mathcal{U}(\hilh)$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\cir$ is $n$ times continuously Fréchet $\Sp^p$-differentiable at every point in $\mathcal{U}(\hilh)$ if and only if $f\in C^n(\cir)$. Take a function $U :\R\rightarrow\mathcal{U}(\hilh)$ such that the function $t\in\R\mapsto U(t)-U(0)$ takes values in $\Sp^{p}$ and is $n$ times continuously $\Sp^{p}$-differentiable on $\R$. Consequently, for $f\in C^n(\cir)$, we prove that $f$ is $n$ times continuously Gâteaux $\mathcal{S}^p$-differentiable at $U(t)$. We provide explicit expressions for both types of derivatives of $f$ in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the $n$th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and $\Sp^{p}$-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev and Tomskova.

math.FA↗

arXiv2024

Functions of unitaries with $\mathcal{S}^p$-perturbations for non continuously differentiable functions

Consider a function $f : \mathbb{T} \to \mathbb{C}$, $n$-times differentiable on $\mathbb{T}$ and such that its $n$th derivative $f^{(n)}$ is bounded but not necessarily continuous. Let $U : \mathbb{R} \to \mathcal{U}(\mathcal{H})$ be a function taking values in the set of unitary operators on some separable Hilbert space $\mathcal{H}$. Let $1<p<\infty$ and let $\mathcal{S}^p(\mathcal{H})$ be the Schatten class of order $p$ on $\mathcal{H}$. If $\tilde{U}:t\in\mathbb{R} \mapsto U(t)-U(0)$ is $n$-times $\mathcal{S}^p$-differentiable on $\mathbb{R}$, we show that the operator valued function $φ: t\in \mathbb{R} \mapsto f(U(t)) - f(U(0)) \in \mathcal{S}^p(\mathcal{H})$ is $n$-times differentiable on $\mathbb{R}$ as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of $φ$ are given in terms of multiple operator integrals and a formula and $\mathcal{S}^p$-estimates for the Taylor remainders of $φ$ are provided.

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arXiv2023

A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces

Fix a metric space $M$ and let $\mathrm{Lip}_0(M)$ be the Banach space of complex-valued Lipschitz functions defined on $M$. A weighted composition operator on $\mathrm{Lip}_0(M)$ is an operator of the kind $wC_f : g \mapsto w \cdot g \circ f$, where $w : M \to \mathbb C$ and $f: M \to M$ are any map. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator $w\widehat{f}$ acting on the Lipschitz-free space $\mathcal F(M)$. In this note, we study the spectrum of such operators, with a special emphasize when they are compact. Notably, we obtain a precise description in the non-weighted $w \equiv 1$ case: the spectrum is finite and made of roots of unity.

math.FA↗

arXiv2023

A pre-adjoint approach on weighted composition operators between spaces of Lipschitz functions

We consider weighted composition operators, that is operators of the type $g \mapsto w \cdot g \circ f$, acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.

math.FA↗

arXiv2021

Compact and weakly compact Lipschitz operators

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat{f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat{f}$ is compact if and only if it is weakly compact.

math.FA↗

arXiv2020

On the dynamics of Lipschitz operators

By the linearization property of Lipschitz-free spaces, any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps $f : M \to M$ and the linear dynamics of their extensions $\widehat{f} : \mathcal F(M) \to \mathcal F(M)$. This not only allows us to relate topological dynamical systems to linear dynamical systems but also provide a new class of hypercyclic operators acting on Lipschitz-free spaces.

math.FA↗

arXiv2020

When do triple operator integrals take value in the trace class?

Consider three normal operators $A,B,C$ on separable Hilbert space $\H$ as well as scalar-valued spectral measures $λ_A$ on $σ(A)$, $λ_B$ on $σ(B)$ and $λ_C$ on $σ(C)$. For any $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ and any $X,Y\in S^2(\H)$, the space of Hilbert-Schmidt operators on $\H$, we provide a general definition of a triple operator integral $Γ^{A,B,C}(ϕ)(X,Y)$ belonging to $S^2(\H)$ in such a way that $Γ^{A,B,C}(ϕ)$ belongs to the space $B_2(S^2(\H)\times S^2(\H), S^2(\H))$ of bounded bilinear operators on $S^2(\H)$, and the resulting mapping $Γ^{A,B,C}\colon L^\infty(λ_A\times λ_B\times λ_C) \to B_2(S^2(\H)\times S^2(\H), S^2(\H))$ is a $w^*$-continuous isometry. Then we show that a function $ϕ\in L^\infty(λ_A\times λ_B\times λ_C)$ has the property that $Γ^{A,B,C}(ϕ)$ maps $S^2(\H)\times S^2(\H)$ into $S^1(\H)$, the space of trace class operators on $\H$, if and only if it has the following factorization property: there exist a Hilbert space $H$ and two functions $a\in L^{\infty}(λ_A \times λ_B ; H)$ and $b\in L^{\infty}(λ_B\times λ_C ; H)$ such that $ϕ(t_1,t_2,t_3)= \left\langle a(t_1,t_2),b(t_2,t_3) \right\rangle$ for a.e. $(t_1,t_2,t_3) \in σ(A) \times σ(B) \times σ(C).$ This is a bilinear version of Peller's Theorem characterizing double operator integral mappings $S^1(\H)\to S^1(\H)$. In passing we show that for any separable Banach spaces $E,F$, any $w^*$-measurable esssentially bounded function valued in the Banach space $Γ_2(E,F^*)$ of operators from $E$ into $F^*$ factoring through Hilbert space admits a $w^*$-measurable Hilbert space factorization.

math.FA↗

arXiv2019

Complete boundedness of multiple operator integrals

In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers.

math.FA↗

arXiv2019

Perturbation theory and higher order $\mathcal{S}^p$-differentiability of operator functions

We establish, for $1 < p < \infty$, higher order $\mathcal{S}^p$-differentiability results of the function $φ: t\in \mathbb{R} \mapsto f(A+tK) - f(A)$ for selfadjoint operators $A$ and $K$ on a separable Hilbert space $\mathcal{H}$ with $K$ element of the Schatten class $\mathcal{S}^p(\mathcal{H})$ and $f$ $n$-times differentiable on $\mathbb{R}$. We prove that if either $A$ and $f^{(n)}$ are bounded or $f^{(i)}, 1 \leq i \leq n$ are bounded, $φ$ is $n$-times differentiable on $\mathbb{R}$ in the $\mathcal{S}^p$-norm with bounded $n$th derivative. If $f\in C^n(\mathbb{R})$ with bounded $f^{(n)}$, we prove that $φ$ is $n$-times continuously differentiable on $\mathbb{R}$. We give explicit formulas for the derivatives of $φ$, in terms of multiple operator integrals. As for application, we establish a formula and $\mathcal{S}^p$-estimates for operator Taylor remainders for a more extensive class of functions. These results are the $n$th order analogue of the results of \cite{KPSS}. They also extend the results of \cite{CLSS} from $\mathcal{S}^2(\mathcal{H})$ to $\mathcal{S}^p(\mathcal{H})$ and the results of \cite{LMS} from $n$-times continuously differentiable functions to $n$-times differentiable functions $f$.

math.FA↗

arXiv2018

Higher order $\Sc^2$-differentiability and application to Koplienko trace formula

Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\in C^n(\mathbb{R})$. We establish that $φ(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\ldots,n$, are bounded. As an application of the second order $\Sc^2$-differentiability, we extend the Koplienko trace formula from the Besov class $B_{\infty1}^2(\R)$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.

math.FA↗

arXiv2017

Schur multipliers on $\mathcal{B}(L^p,L^q)$

Let $(Ω_1, \mathcal{F}_1, μ_1)$ and $(Ω_2, \mathcal{F}_2, μ_2)$ be two measure spaces and let $1 \leq p,q \leq +\infty$. We give a definition of Schur multipliers on $\mathcal{B}(L^p(Ω_1), L^q(Ω_2))$ which extends the definition of classical Schur multipliers on $\mathcal{B}(\ell_p,\ell_q)$. Our main result is a characterization of Schur multipliers in the case $1\leq q \leq p \leq +\infty$. When $1 < q \leq p < +\infty$, $ϕ\in L^{\infty}(Ω_1 \times Ω_2)$ is a Schur multiplier on $\mathcal{B}(L^p(Ω_1), L^q(Ω_2))$ if and only if there are a measure space (a probability space when $p\neq q$) $(Ω,μ)$, $a\in L^{\infty}(μ_1, L^{p}(μ))$ and $b\in L^{\infty}(μ_2, L^{q'}(μ))$ such that, for almost every $(s,t) \in Ω_1 \times Ω_2$, $$ϕ(s,t)=\left\langle a(s), b(t) \right\rangle.$$ Here, $L^{\infty}(μ_1, L^{r}(μ))$ denotes the Bochner space on $Ω_1$ valued in $L^r(μ)$. This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on $\mathcal{B}(\ell_p,\ell_q)$.

math.FA↗

arXiv2015

Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case

We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.

math.FA↗

arXiv2015

Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$

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