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Continuous Linear and Bilinear Schur Multipliers and Applications to Perturbation Theory Clément Coine

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Continuous Linear and Bilinear Schur Multipliers and Applications to Perturbation Theory Clément Coine Continuous linear and bilinear Schur multipliers and applications to perturbation theory Clément Coine To cite this version: Clément Coine. Continuous linear and bilinear Schur multipliers and applications to perturbation theory. Mathematics [math]. Université Bourgogne-Franche-Comté, 2017. English. tel-02177442 HAL Id: tel-02177442 https://tel.archives-ouvertes.fr/tel-02177442 Submitted on 9 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. École doctorale Carnot-Pasteur THÈSE DE DOCTORAT présentée par CLÉMENT COINE Pour obtenir le grade de Docteur de Mathématiques de l’Université Bourgogne Franche-Comté Continuous linear and bilinear Schur multipliers and applications to perturbation theory Thèse soutenue le 30 Juin 2017, devant le jury composé de : Isabelle CHALENDAR Examinateur Christian LE MERDY Directeur de thèse Eric RICARD Rapporteur Anna SKRIPKA Rapporteur Ivan TODOROV Examinateur Quanhua XU Examinateur et Président du jury ii iii REMERCIEMENTS Je tiens tout d’abord à remercier Christian Le Merdy pour m’avoir initié au monde de la recherche. Il a été très présent durant ces trois années et m’a appris énormément de choses, toujours avec pédagogie. Il a partagé avec moi ses connaissances et de nom- breux conseils sur le métier de chercheur. Il fut une grande source d’inspiration et le sera encore dans le futur. Je voudrais ensuite remercier Isabelle Chalendar, Eric Ricard, Anna Skripka, Ivan Todorov et Quanhua Xu pour avoir accepté de faire partie de mon jury. Ça a été pour moi un grand honneur et un plaisir de partager mon travail avec eux. Je remercie tout particulièrement Anna et Eric pour leur travail de rapporteur. Une partie de cette thèse a été réalisée en collaboration avec Denis Potapov, Anna Skripka, Fedor Sukochev et Anna Tomskova. Je les remercie d’avoir partagé ce travail avec moi. Je remercie tout particulièrement Anna Tomskova et Fedor Sukochev pour leurs visites en France durant lesquelles ils m’ont initié aux problèmes de Peller. Le travail de cette thèse a été très influencé par cette question et leur connaissance du su- jet a permis une collaboration fructueuse. J’ai eu la chance durant cette thèse d’effectuer un séjour de recherche de deux mois à Albuquerque. Je remercie chaleureusement Anna Skripka pour son accueil et pour avoir échangé ses connaissances avec moi. Nous avons pu discuter sur différentes questions mathématiques très intéressantes et cette collaboration a abouti sur l’écriture d’un article. Je remercie également Arup pour m’avoir fait visité une partie de la ville et pour les conversations que nous avons eues autour des succulents mets mexicains. Merci au Laboratoire de Mathématiques de Besançon de m’avoir accueilli et aux différents membres qui ont toujours été présents en cas de problème ou de question. Je tiens également à remercier l’équipe d’analyse fonctionnelle au sein de laquelle j’ai évolué et qui sait créer un cadre de travail très stimulant. Je remercie tous les excellents professeurs que j’ai rencontrés au cours de ma scolar- ité et qui ont renforcé mon goût pour les mathématiques, notamment les professeurs d’analyse qui m’ont toujours inspiré. iv Merci à toute l’équipe de doctorants, post-docs et ATER pour l’ambiance et les bons moments passés ensemble. Merci au bureau 401B pour avoir crée un climat de travail très agréable. Je remercie particulièrement Colin pour les fous rires et pour avoir tou- jours une bouteille d’eau sur lui, Marine pour les 8 années passées ensemble ainsi que pour la décoration du bureau et Olga pour son amitié et les nombreuses heures passées à discuter de tout et de rien. Merci également à Aude, Charlotte et Cyril qui ont été présents pour m’aider pendant ces trois années. Je remercie également tous les autres doctorants et amis du qui ont fait partie du labo : Alessandro, Haonan, Isabelle, Jo- hann, Lucie, Michael, Michel, Othman, Pammella, Quentin, Runlian, Sabrina, Youssef. Je remercie également Eva pour les discussions autour de crêpes, et Kamilia pour son amitié et les très bons moments passés lors des conférences. Enfin, je veux remercier tous mes proches, mes amis et ma famille, qui m’ont permis de faire cette thèse dans les meilleures conditions. Un merci tout particulier à mes parents, mes sœurs et mon grand-père pour avoir été très présents tout au long de mes études et pour m’avoir encouragé lors de ces trois années de thèse. Merci également aux petits nouveaux, Julia et Lucas, pour le bonheur qu’ils procurent. v Contents Introduction1 1 Résumé de la thèse...............................1 2 Summary of the thesis.............................4 3 Notations.....................................6 1 Preliminaries9 1.1 Tensor products.................................9 1.1.1 Projective and injective tensor product................9 1.1.2 Lapresté norms............................. 11 1.1.3 Haagerup tensor product....................... 12 1.1.4 Dual norm................................ 15 1.1.5 (p; q)−Factorable operators....................... 15 1.2 Schatten classes................................. 18 1.2.1 Definition and duality......................... 18 1.2.2 Tensor products of Hilbert space operators and trace duality... 19 p 1.3 Lσ-spaces and duality............................. 22 1 ∗ 1.4 Measurable factorization in Lσ (Ω; Γ2(E; F )) ................ 25 1.4.1 The main result............................. 25 1.4.2 A special case.............................. 32 2 Linear Schur multipliers 35 2.1 Classical Schur multipliers........................... 35 2.2 Continuous Schur multipliers on B(L2) ................... 36 2.3 Schur multipliers on B(Lp;Lq) ......................... 38 2.3.1 Definition and connection with the classical Schur multipliers.. 38 2.3.2 Schur multipliers and factorization.................. 42 2.3.3 An application : the main triangle projection............ 48 2.4 Inclusion theorems............................... 50 2.5 Perspectives................................... 54 3 Bilinear Schur multipliers 57 3.1 Definition and notations............................ 57 3.2 Bilinear Schur multipliers valued in S1 .................... 58 3.3 Continuous bilinear Schur multipliers.................... 61 3.3.1 Definition................................ 62 3.3.2 S1-boundedness............................ 64 3.4 Perspectives................................... 68 vi 4 Multiple Operator Integrals 69 4.1 Definition.................................... 70 4.1.1 Normal operators and scalar-valued spectral measures...... 70 4.1.2 Multiple operator integrals associated with operators....... 71 4.1.3 Passing from operators to functions................. 77 4.2 Finite dimensional case............................. 79 4.2.1 Double operator integrals....................... 79 4.2.2 Triple operator integrals........................ 81 4.3 Characterization of S2 × S2 !S1 boundedness............... 82 4.4 Complete boundedness of triple operator integrals............. 87 4.5 Perspectives................................... 92 5 Resolution of Peller’s problems 93 5.1 Statement of the problems........................... 93 5.2 Perturbation theory for selfadjoint operators................ 95 5.2.1 Approximation in multiple operator integrals........... 97 5.2.2 Proof of the main result........................ 100 5.2.3 Connection with Peller’s problem.................. 106 5.3 The self-adjoint case.............................. 107 5.3.1 A few properties of triple operator integrals............ 107 5.3.2 Finite-dimensional constructions................... 111 5.3.3 A solution to Peller’s problem for selfadjoint operators...... 121 5.4 The unitary case................................. 122 5.4.1 Preliminary results........................... 122 5.4.2 Finite-dimensional constructions................... 126 5.4.3 A solution to Peller’s problem for unitary operators........ 130 5.5 Perspectives................................... 134 Bibliography 135 1 Introduction 1 Résumé de la thèse Le travail de cette thèse a été dans un premier temps motivé par la résolution du prob- lème de Peller concernant la formule de trace de Koplienko-Neidhardt. Celui-ci est en lien avec les perturbations du second ordre pour le calcul fonctionnel. En effet, le problème était de déterminer, pour une fonction f 2 C2(R) dont la dérivée seconde est bornée, et pour deux opérateurs autoadjoints A et K sur un espace de Hilbert séparable H tels que K 2 S2(H) est un opérateur de Hilbert-Schmidt, si l’opérateur d f(A + K) − f(A) − f(A + tK) (1) dt t=0 appartient à l’espace S1(H) des opérateurs à trace. Cette question a été soulevée par V. Peller dans [Pel05], où il a également conjecturé que la réponse à cette question était négative. Pour résoudre ce problème, il est important de comprendre tout d’abord dans quels cas l’opérateur (1) est bien défini. Lorsque A est borné ou quand f a une dérivée bornée, l’opérateur est bien défini et appartient à S2(H). Sinon, le sens de (1) n’est pas clair, mis à part dans certains cas particuliers. Peller a par exemple défini dans [Pel05] 2 l’opérateur (1) par approximation
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