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Isabelle Chalendar

Publications and source records attributed to Isabelle Chalendar.

At least 19 recordsLinked to original sources
arXiv2025

On the relation between distances and seminorms on Fr\'echet spaces, with application to isometries

A study is made of linear isometries on Fr\'echet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach--Stone theorem to surjective and nonsurjective cases.

math.FA↗
arXiv2024

Embedding of some classes of operators into strongly continuous semigroups

In this paper we study the embedding problem of an operator into a strongly continuous semigroup. We obtain characterizations for some classes of operators, namely composition operators and analytic Toeplitz operators on the Hardy space H^2. In particular, we focus on the isometric ones using the necessary and sufficient condition observed by T. Eisner.

math.FA↗
arXiv2024

Linear isometries on the annulus: description and spectral properties

We give a complete characterisation of the linear isometries of ${\rm Hol}(\Omega)$, where $\Omega$ is the half-plane, the complex plane or an annulus centered at 0 and symmetric to the unit circle. Moreover, we introduce new techniques to describe the holomorphic maps on the annulus that preserve the unit circle, and we finish by proving results about the spectra of the linear isometries on the annulus.

math.FA↗
arXiv2024

Eventual Ideal Properties of the Riemann-Liouville Analytic Semigroup

In this paper, we revisit the Riemann--Liouville analytic semigroup. In particular, we completely characterize the membership to the Schatten class $S^r$ on $L^2(0,1)$, as well as the membership to the class of nuclear operators on $L^p(0,1)$, $p\geq 1$, and the membership to the ideal of absolutely $r$-summing operators for any $r\geq 1$.

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arXiv2023

A note on composition operators on model spaces

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $ϕ\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $Ψ$ satisfying the functional equation $Ψ\circϕ=τΨ$, where $τ$ is a unimodular constant. We prove that this problem has a solution if and only if $ϕ$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $τ=1$. If in addition, $ϕ$ is parabolic, we prove that the problem has a solution $Ψ$ for $any$ unimodular $τ$. Finally, we show that if $ϕ$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $τ$ satisfy $f\circϕ=τf$.

math.CV↗
arXiv2023

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

We first give a general error estimate for the nonconforming approximation of a problem for which a Banach-Ne{\v c}as-Babu{\v s}ka (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $θ$--scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $θ$ $\le$ 1 2. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

math.NA↗
arXiv2022

Extensions of derivations and symmetric operators

Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $Φ$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $Φ$ is a unitary operator. As corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from Hto H +. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).

math.NA↗
arXiv2022

Lions' representation theorem and applications

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.

math.FA↗
arXiv2020

Galerkin approximation of linear problems in Banach and Hilbert spaces

In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V' a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions are taken in a finite dimensional subspace of V. We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that U has a finite dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case, a generalization of the Aubin-Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side L, as shown by several applications. Finally, a section entitled "Supplement" provides some consequences of our results for the approximation of saddle point problems.

math.NA↗
arXiv2019

In Koenigs' footsteps: Diagonalization of composition operators

Let $φ:\mathbb{D} \to \mathbb{D}$ be a holomorphic map with a fixed point $α\in\mathbb{D}$ such that $0\leq |φ'(α)|<1$. We show that the spectrum of the composition operator $C_φ$ on the Fréchet space $ \textrm{Hol}(\mathbb{D})$ is $\{0\}\cup \{ φ'(α)^n:n=0,1,\cdots\}$ and its essential spectrum is reduced to $\{0\}$. This contrasts the situation where a restriction of $C_φ$ to Banach spaces such as $H^2(\mathbb{D})$ is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schröder symbol on arbitrary Banach spaces of holomorphic functions.

math.SP↗
arXiv2019

The reproducing kernel thesis for lower bounds of weighted composition operators

It is shown that the property of being bounded below (having closed range) of weighted composition operators on Hardy and Bergman spaces can be tested by their action on a set of simple test functions, including reproducing kernels. The methods used in the analysis are based on the theory of reverse Carleson embeddings.

math.FA↗
arXiv2016

Subspaces of $\displaystyle H^{p}$ linearly homeomorphic to $l^{p}.$

We present two fast constructions of weak*-copies of $\ell ^\infty$ in $H^{\infty}$ and show that such copies are necessarily weak*-complemented. Moreover, via a Paley-Wiener type of stability theorem for bases, a connection can be made in some cases between the two types of construction, via interpolating sequences (in fact these are at the basis of the second construction). Our approach has natural generalizations where H $\infty$ is replaced by an arbitrary dual space and $\ell ^{\infty }$ by $\ell^{p}$ (1 $\le$ p $\le$ $\infty$) relying on the notions of generalized interpolating sequence and bounded linear extension. An old (very simple but unpublished so far) construction of bases which are Besselian but not Hilbertian finds a natural place in this development.

math.CV↗
arXiv2016

Recent results on truncated Toeplitz operators

Truncated Toeplitz operators are compressions of Toeplitz operators on model spaces; they have received much attention in the last years. This survey article presents several recent results, which relate boundedness, compactness, and spectra of these operators to properties of their symbols. We also connect these facts with properties of the natural embedding measures associated to these operators.

math.FA↗
arXiv2015

Inner functions and operator theory

This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation.

math.FA↗
arXiv2014

An extremal problem for characteristic functions

Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset L^\infty$ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $E$ to $z^nH^\infty$. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

math.CV↗