## Séminaire d’Analyse fonctionnelle 2017-2018

par - publié le

Vous trouverez ci-dessous le planning du séminaire d’Analyse
Fonctionnelle pour l’année 2017-2018. Le programme de l’année en cours se trouve
ici.

Pour contacter la responsable (Yulia Kuznetsova) : yulia.kuznetsova@univ-fcomte.fr.
Pour s’abonner au séminaire : ACM.

### Mai

-Mercredi 2 mai à 13h45: Sean Harris, Canberra, Australia.
Weyl Pseudodifferential Operators in Ornstein-Uhlenbeck Settings

The classical Weyl pseudodifferential calculus is a particular choice of quantisation'' - a way to take functions of the position and momentum operators on $R^n$. This pseudodifferential calculus allows study of complicated operators to be (mostly) encapsulated by studying their symbols. Ornstein-Uhlenbeck (OU) operators are analogs of the Laplacian adapted to spaces with Gaussian measure, and arise in many areas including stochastic analysis, quantum field theory and harmonic analysis. They are particularly nasty if tackled analytically and directly, with very rigid structures (for example, the standard OU operator has only $H^\infty$ calculus on $L^p$, the proof of which takes over 100 pages in full detail !). From one of these origins, the OU operator arises naturally as a function'' of position- and momentum-like operators, which suggests that the ideas of Weyl calculi may be applicable. After explaining these and other important concepts, I will explain my current work in adapting the Weyl pseudodifferential calculus to the OU setting. This is of a very different flavour to the standard Weyl pseudodifferential calculus, with interesting links to complex analysis and Banach algebras. At present, it seems that using the Weyl calculus splits the problem of studying OU operators into an algebraic part and an analytic part, the analytic part being almost trivial when compared to the analysis used for studying OU directly.

-Mardi 15 mai: Uwe Franz, LMB.

-Mardi 22 mai: Matias Raja, Universidad de Murcia.

-Lundi, 28 mai à ciser: Valentin Ferenczi, Universidade de Sao Paulo.
Ultrahomogeneïté et propriété de Ramsey dans les espaces $L_p$

-Mardi 29 mai: à 13:30 --- 1 juin 2018 à 12:30, .
Ecole de printemps 2018 du GdR AFHP

Guillaume Aubrun (Lyon), Marek Cúth (Prague) et Sophie Grivaux (Lille). Programme: https://trimestres-lmb.univ-fcomte.fr/Spring-school-2018-of-GdR-AFHP?lang=fr

### Avril

-Mardi 3 avril: Rauan Akylzhanov, Imperial College London.
Smooth dense subalgebras and Fourier multipliers on compact quantum groups

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded self-adjoint Dirac-like operator with compact resolvent. Further, we characterise the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. Further, we show that composition of two regular pseudo-differential operators is a regular pseudo-differential operator. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for Lp - Lq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to the proposed smooth structure. We check explicitly that these conditions hold true on the quantum SUq(2) for both its 3-dimensional and 4-dimensional calculi. Joint work with Michael Ruzhansky and Shahn Majid.

-Mardi 24 avril: Yi-Jun YAO, University of Fudan.
Sur un lemme de Connes
Nous allons presenter les détailles concernant le Lemme VI.3.9 du livre "Noncommutative Geometry" d'Alain Connes.

### Mars

-Mardi 13 mars: Anna Skripka, University of New Mexico.
Schur multipliers in perturbation theory

We will recall classical Schur multipliers acting on matrices and consider their generalizations to multilinear transformations arising in infinite dimensional perturbation theory. As an application, we will discuss several recent results on approximation of operator functions.

-Mardi 20 mars: Bruno de Mendoça Braga , York University, Toronto.
Nonlinear weakly sequentially continuous embeddings between Banach spaces

In this talk, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $\| e_1+\dots + e_k \|_\delta \lesssim \| e_1+\dots+e_k\|_S$, where $\delta$ is the modulus of asymptotic uniform convexity of $Y$.

-26-27 mars: Journées en Analyse Fonctionnelle, .

Lundi 26 mars: salle 316B
14:00 Isabelle Chalendar (Marne-La-Vallée)
15:00 Julien Bichon (Clermont-Ferrand)
16:00 coffee break
16:30 Karim Kellay (Bordeaux)

Mardi 27 mars
10:00 Yulia Kuznetsova, soutenance HDR (LMB): amphi A

Salle 316 B:
14:00 Alexander Borichev (Marseille)
15:00 Éric Ricard (Caen)
16:00 coffe break
16:30 Pierre Fima (Paris 7)

### Février

-Mardi 6 février: Safoura Zadeh, IMPAN (Varsovie).
Markov's problem for k-absolutely monotone polynomials
Studying the specific gravity of a solution as a function of the percentage of the dissolved substance D. Mendeleev came across a problem that, in mathematical term, asks how large $\lvert p'(x)\rvert$ on $[-1,1]$ can be if $p(x)$ is a quadratic polynomial with $\lvert p(x)\rvert\leq 1$ on $[-1,1]$. Mendeleev showed that $\lvert p'(x)\rvert\leq 4$ and talked to A. A. Markov about the problem. Markov found the question fascinating and studied the problem for a real polynomial of degree at most n. He proved that $\lvert p'(x)\rvert \leq n^2$ on $[-1,1]$ when $\lvert p(x)\rvert\leq 1$ on $[-1,1]$. Later Markov's brother V. A. Markov studied the problem for the $m$-th derivative, when $m\lt n$. In this talk we study Markov's problem for $m$-th derivative of a family of polynomials called k-absolutely monotone polynomials. As a consequence, we obtain a sharp version of Bernstein inequality for monotone polynomials as well as a new simple proof of Markov's inequality for monotone polynomials. This is based on a joint work with Oleksiy Klurman.

-Mardi 13 février: Kangwei Li, Basque Center for Applied Mathematics (Bilbao).
Extrapolation for multilinear Muckenhoupt class of weights and applications

In this talk, I will introduce our recent progress on extrapolation theory. In the linear case, the extrapolation theory is well understood. However, in the multilinear case, the extrapolation was only known for product $A_p$ weights. The multilinear $A_{P}$ weight, which was introduced in 2009, no extrapolation theory was known before. In this talk, I will give a full solution to this problem. As applications, we can improve the weighted estimates for the bilinear Hilbert transform, the multilinear Marcinkiewicz-Zygmund inequality etc. This talk is based on joint work with José María Martell and Sheldy Ombrosi.

-Mardi 27 février: Gaspar Mora, Universidad de Alicante.
On the distribution of the zeros of exponential polynomials

In this talk we analyse the distribution of the zeros of exponential polynomials $$h(z) :=1+\sum_{k=1}^N a_k e^{-zr_k}; \ \ z, a_k\in \mathbb C, \ r_k>0, \ N\geq 1$$ by means of the structure of the set $R_h(z) :=\overline{\{ \Re z:h(z)=0\}}$. Our special interest is focused when $h(z)$ is a partial sum of the Riemann zeta function or a partial sum of the Dirichlet alternating series.

### Janvier

-Mardi 16 janvier: Un Cig Ji, Chungbuk National University, Korea.
Inequalities for Positive Module Operators on von Neumann Algebras

We establish the Cauchy-Schwarz and Golden-Thompson inequalities for module operators, a generalization of a (noncommutative) conditional expectation, on a von Neumann algebra. We apply these inequalities to the Bennett inequality and a uncertainty relation, a generalization of the Schrödinger uncertainty relation, for conditional expectations. This is a joint work with B. J. Choi and Y. Lim.

### Décembre

-Mardi 12 décembre: Philipp Varso, Greifswald.
Central Limit Theorem for General Universal Products

To model independence in quantum probability theory one uses so-called universal products, which are in general described by tensor categories of algebraic quantum probability spaces $(A,\varphi )$. In [1] Muraki has shown how to classify such products and in particular obtained that only -five universal products exist. But there are examples which do not fit into Muraki's framework, for instance if one wants to deal with a tuple of linear functionals $\varphi ^(i)$ on the algebra $A$ like it has been done in the case of $c$-freeness by Bozejko and Speicher in [2].The case of bi-freeness by Voiculescu [3], where in particular the underlying algebra $A$ is isomorphic to the free product of two algebras $A^(1)$ and $A^(2)$, is also not covered by Muraki's classi-cation. In [4] Schürmann and Manzel present a unified approach to cumulants, which includes the above concepts of independence. This is achieved by considering a certain category of algebraic non-commutative probability spaces, denoted by $algP_d,m$, which consist of an $m$-tuple of subalgebras and a $d$-tuple of linear functionals and therefore allows to investigate $(d,m)$-independence induced by a so-called u.a.u.-product.The first noncommutative version of a central limit theorem dates back to von Waldenfels [5]. In this talk we want to present a noncommutative version of a central limit theorem for a u.a.u.-product in $algP_d,m$, where we make use of the so-called Lachs functor [6], which operates between certain tensor categories. References: [1] Muraki, Naofumi : The five independences as natural products. In-n. Dimens. Anal. Quantum Probab. Relat. Top. 6.3 (2003), 337-371. [2] Bozejko, Marek ; Speicher, Roland : $\psi$ -independent and symmetrized white noises. In : Quantum probability \& related topics. QP-PQ, VI, 219-236. World Sci. Publ., River Edge, NJ, 1991. [3] Voiculescu, Dan-Virgil : Free probability for pairs of faces I. Comm. Math. Phys. 332.3 (2014), 955-980. [4] Manzel, Sarah ; Schürmann, Michael : Non-Commutative Stochastic Independence and Cumulants. Preprint arXiv:1601.06779 (2016), 42 pages. To appear in IDAQP 20.2 (2017). [5] von Waldenfels, W. : An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42.2 (1978), 135-140. [6] Lachs, Stephanie : A New Family of Universal Products and Aspects of a Non-Positive Quantum Probability Theory. PhD thesis. Ernst-Moritz-Arndt-Universität Greifswald, 2015.

-Mardi 19 décembre: Ignacio Vergara, ENS Lyon.

Les multiplicateurs de Schur sont des fonctions à deux variables sur un ensemble $X$ qui définissent des opérateurs bornés sur $B(\ell _2(X))$ par multiplication des coefficients matriciels. Lorsque l'ensemble $X$ est un graphe, on peut étudier le cas particulier des multiplicateurs radiaux, c'est-à-dire, des fonctions que ne dépendent que de la distance entre chaque paire de sommets. Les multiplicateurs radiaux sur un arbre homogène ont été caractérisés par Haagerup, Steenstrup et Szwarc en termes de certaines matrices de Hankel. Dans cet exposé, je présenterai des extensions de ce résultat à des produits d'arbres, des produits de graphes hyperboliques et aux complexes cubiques CAT(0) de dimension finie.

### Novembre

-Mardi 21 novembre: Safoura Zadeh, IMPAN, Warsaw.
Isometric algebra isomorphisms between weighted $L^p$-algebras

In Chapter 11 of his renowned book "Théorie des opérations linéaires" S. Banach gave a description of linear norm preserving operators on the spaces $L^p$ and $l^p$, $1\leq p<\infty,\ p\not =2$. The proofs are not stated completely and the theorems are not given in their full generality. This was fulfilled by J. Lamperti who provided new proofs for a more general theorem ; besides being set in arbitrary ($\sigma$-finite) measure spaces, Lamperti's result accepts values $p<1$. Later, Parrot and Strichartz independently extended Lamperti's result to convolution $L^p$-algebras. They showed that if $G$ and $H$ are compact topological groups and if $T:L^p(G)\to L^p(H)$, $1\leq p<\infty,\ p\not =2$, is an isometric algebra isomorphism then there is an isomorphisms of topological groups $\phi :G\to H$, a continuous character $\gamma :G\to (0,+\infty )$ and a constant $c$ such that $$Tf(y)=c\gamma \circ \phi ^-1(y) f\circ \phi ^-1(y)\ \ \ (y\in H).$$ In this talk I give a description of isometric algebra isomorphisms between weighted $L^p$-algebras on locally compact groups. This is based on a join work with Yulia Kuznetsova.

-Mardi 28 novembre: Adam Skalski, IMPAN, Warsaw.
Translation invariant Dirichlet forms in the context of locally compact quantum groups

Since the work of Cipriani on one hand and Goldstein and Lindsay on the other in the 1990s it is known that certain natural class of symmetric Markov semigroups on a von Neumann algebra M equipped with a faithful normal state admitsextensions to associated Haagerup $L^p$-spaces and is characterised via a Dirichlet property of the generating quadratic form on the$L^2$-space.Recently Cipriani, Franz and Kula studied a special class of such semigroups associated to compact quantum groups. In this talk I will discuss how their results extend to the framework of locally compact quantum groups, where two new important technical features appear : there is no natural `algebraic' domain for the generator and one needs to work with weights, as opposed to finite states (using the appropriate Dirichlet form result provided by Goldstein and Lindsay). I will also present some applications of Dirichlet forms to the study of geometric properties of quantum groups.This is based on the joint work with A.Viselter.

### Octobre

-Mardi 10 octobre à 14h00: Runlian Xia, LMB.
Les espaces de Hardy locaux à valeur opératorielle et les applications sur les opérateurs pseudo-différentiels

La soutenance de thèse

-Mardi 17 octobre: Takahiro Hasebe, Hokkaido University.
Séminaire d'Analyse Fonctionnelle

-Jeudi 19 octobre à 11h00: Ping Zhong, University of Waterloo and Wuhan University.
Some noncommutative probability aspects of meandric systems

The talk will consider a family of diagrammatic objects (well-known to combinatorialists and mathematical physicists) which go under the names of "meandric systems" or "semi-meandric systems". I will review some connections which these objects are knownto have with free probability, and I will show in particular how the so-called "semi-meandric polynomials" can be retrieved from a natural consideration of operators on the $q$-Fock space. This is joint work with Alexandru Nica.

-Mardi 24 janvier: Srinivasan Raman, .
$E_0-$semigroups on factors

I will review the current progress on endomorphism semigroups on factors, particularly on non-type-I factors. This is mostly my joint work with Oliver T Margetts.

### Septembre

- Mardi 26 septembre 2017: Colin Petitjean, UFC.
The linear structure of some dual Lipschitz free spaces

Consider a metric space $M$ with a distinguished point $0_M$. Let $Lip_0(M)$ be the Banach space of Lipschitz functions from $M$ to $\mathbb R$ satisfying $f(0_M) = 0$ (the canonical norm being the best Lipschitz constant). The Lipschitz-free space $\mathcal F(M)$ over $M$ is defined as the closed linear span in $Lip_0(M)^*$ of $\delta(M)$ where $\delta (x)$ denotes the Dirac measure defined by $\langle \delta (x) , f \rangle = f(x)$. The Lipschitz free space $\mathcal F(M)$ is a Banach space such that every Lipschitz function on $M$ admits a canonical linear extension defined on $\mathcal F(M)$. It follows easily from this fundamental linearisation property that the dual of $\mathcal F (M)$ is in fact $Lip_0(M)$. A considerable effort to study the linear structure and geometry of these spaces has been undergone by many researchers in the last two or three decades. In this talk, we first focus on some classes of metric spaces $M$ for which $\mathcal F(M)$ is isometrically isomorphic to a dual Banach space. After a quick overview of the already known results in this line, we define and study the notion of "natural predual". A natural predual is a Banach space $X$ such that $X^* = \mathcal F(M)$ isometrically and $\delta(M)$ is $\sigma(\mathcal F(M),X)$-closed. As we shall see, $\delta(M)$ is always $\sigma(\mathcal F(M),Lip_0(M))$-closed but it may happened that it is not $\sigma(\mathcal F(M),X)$-closed for some predual $X$. We characterise the existence of a natural predual in some particular classes of metric spaces. Notably, we concentrate on the class of uniformly discrete and bounded (shortened u.d.b.) metric spaces, for which it is well known that $\mathcal F(M)$ is isomorphic to $\ell_1$. In particular, we exhibit an example of a u.d.b. metric space $M$ for which $\mathcal F(M)$ is a dual isometrically but which does not have any natural predual. We also provide a u.d.b. metric space $M$ such that $\mathcal F(M)$ is not a dual isometrically. We finish with the study of the extremal structure of Lipschitz free spaces admitting a natural predual. This is part of a joint work with L. García-Lirola, A. Procházka and A. Rueda Zoca.