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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Sobre os produtos de Stratonovich e de Berezin de símbolos na esfera / About the Stratonovich and Berezin product of symbol in the sphere

Harb, Nazira Hanna 09 May 2014 (has links)
Esta tese versa sobre os produtos de Stratonovich e de Berezin de funções na esfera \'S POT. 2\'. Cada um destes produtos é definido atravéz de uma correspondência de símbolos, que é uma aplicação linear bijetiva entre operadores lineares num espaço de Hilbert complexo de dimensão n + 1, ou seja matrizes complexas (n + 1) × (n + 1), e polinômios complexos de grau próprio n definidos na 2-esfera, PolyC(\'S POT. 2\')n, satisfazendo algumas propriedades básicas, como equivariância pela ação do grupo de rotações SO(3), preservação das estruturas reais e normalização [12]. Mais geralmente, toda correspondência define um produto associativo mas não comutativo em PolyC(\'S POT. 2\')n induzido do produto de operadores, chamado de produto twisted em PolyC(\'S POT. \'2)n. Cada um destes produtos twisted, por sua vez, pode ser escrito na forma integral f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, onde f, g PolyC(\'S POT. 2)n, \'n IND. 1\',\' n IND. 2\', n \'S POT. 2\'. Em tal representação integral, todas as propriedades do produto twisted são convertidas em propriedades do trikernel integral L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' C. Os produtos twisted estudados nesta tese são os produtos induzidos pela correspondência padrão de Stratonovich e a correspondência padrão de Berezin, respectivamente, que num certo limite assintótico 2j = n definem deformações estritas da álgebra de Poisson de \'S POT. 2\' [12]. Para cada um deste dois produtos, denotados por \'n SOB. 1\' e \'n SOB. b\' respectivamente, seu trikernel integral é denotado por L \'SOB. j 1\' e L \'j SOB. ~b, respectivamente. A primeira parte desta tese consistiu em desenvolver fórmulas mais tratáveis para L \'j SOB. 1\' e L j SOB. b\' nos casos de número de spin j = 1/2, 1, 3/2, 2, fórmulas estas escritas em termos de funções de dois e de três pontos, invariantes por SO(3), como produtos escalares e determinantes. Nossa esperança inicial era de que pudéssemos encontar padrões que nos permitissem inferir fórmulas fechadas para cada um destes trikernels, válidas para qualquer j, ou pelo menos que nos permitissem inferir fórmulas assintóticas para estes trikernels quando 2j = n . Porém, o grau de complexidade das fórmulas desenvolvidas se mostrou fortemente crescente com j, frustrando nossas expectativas iniciais. Partimos então para uma exploração preliminar de um tipo de aproximação assintótica destes produtos de certas funções oscilatórias na esfera. Mais precisamente, na segunda parte desta tese, preparamos e estudamos preliminarmente o produto de Stratonovich e o produto de Berezin (assim como o produto pontual) de dois harmônicos esféricos, \'Y POT. m1 INF. l1\' e \'Y POT. m2 INF. l2 PolyC(\'S POT. 2\')n, no limite assintótico quando tanto \'l IND. 1\' como \'l IND. 2\' tendem a infinito linearmente com n (mantendo \'l IND. i\' n). Este tipo de assintótica para tais produtos, que faz parte do que chamamos mais geralmente de high-l asymptotics, difere muito do tipo de assintótica estudada de forma detalhada em [12], na qual n , mas \'l IND. 1\' e \'l IND. 2\' são mantidos finitos. Então, a partir de um exemplo particular para nossa exploração preliminar, levantamos uma conjectura sobre como estes produtos se comparam no limite assintótico quando \'l IND. 1\' e \'l IND. 2\' tendem para infinito linearmente com o número de spin j / This thesis is about the Stratonovich and the Berezin products of functions on the 2-sphere. Each one of these products is defined via a spin-j symbol correspondence, a linear bijective map from the space of operators on an (n + 1)-dimensional complex Hilbert Space, i.e. (n + 1) × (n + 1) complex matrices, and the space of complex polynomials on \'S POT. 2\' of proper degree n, denoted PolyC(\'S POT. 2\')n, satisfying certain basic properties like equivariance under the action of SO(3), preservation of real structures and normalization [12]. More generally, every spin-j symbol correspondence defines an associative noncommutative product on PolyC(\'S POT. 2\')n induced from the operator product, which is called a twisted product on PolyC(\'S POT. 2\')n. Each twisted product can be written in integral form as f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, where f, g PolyC(\'S POT. 2) \' > OR =\' n, \'n IND. 1\',\' n IND. 2\', n \'IT BELONGS\' \'S POT. 2\'. In such and an integral representation, all properties of the twisted product are translated to properties of its integral trikernel L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' \'ARROW\' C. The twisted products studied in this thesis are the ones obtained via the standard Stratonovich-Weyl and the standard Berezin symbol correspondences, respectively, which in a certain asymptotic limit 2j = n \'ARROW\' \'INFINITY\' define strict deformation quantizations of \'S POT. 2\', i.e. strict deformations of the Poisson algebra of \'S POT. 2\' [12]. For 10 each of these two products on PolyC(\'S POT. 2\') \'< OR =\' n, denoted by * \'n SOB. 1\' and \'n SOB. b\' respectively, its integral trikernel is denoted by L\'j SOB. 1\' and L\'j SOB. b\' , respectively. In the first part of this thesis, we obtained better formulae for these trikernels, for values of spin number j = 1/2, 1/3/1, 2. These formulas are written in terms of SO(3)-invariant functions of two and three points on \'S POT. 2\', like scalar products and determinants. We initially hoped to be able to obtain closed formulae for these trikernels which would be valid for every j, or at least be able to infer asymptotic formulas for these trikernels when 2j = n \'ARROW\' \' INFINITY\' . However, the degree of complexity of the formulae we have obtained increases strongly with j, frustrating our initial expectations. We thus started on a preliminary investigation of a kind of asymptotic approximation for these products of certain oscillatory functions on the sphere. More precisely, in the second part of this thesis, we prepared and preliminarily studied the Stratonovich product and the Berezin product (as well as the pointwise product) of spherical harmonics Y \'m1 SOB. l1\' and Y \'m2 SOB. l2\' PolyC(\'S POT. 2\')n, in the asymptotic limit when l1 and l2 tend to infinity linearly with n (keeping \'l IND. i\' \'< OR =\' n). This kind of asymptotics for these products, belonging to what we more generally callhigh-l asymptotics, differs drastically from the kind of asymptotics studied in detail in [12], in which n \'ARROW\' \'INFINITY\' but \'l IND. 1\' and \'l IND. 2\' are kept finite. Then, based on a particular example of our preliminary exploration, we advanced a conjecture on how these products behave and compare with each other, in the asymptotic limit when \'l IND. \'1\' and \'l IND. 2\' tend to infinity linearly with the spin number j
2

Sobre os produtos de Stratonovich e de Berezin de símbolos na esfera / About the Stratonovich and Berezin product of symbol in the sphere

Nazira Hanna Harb 09 May 2014 (has links)
Esta tese versa sobre os produtos de Stratonovich e de Berezin de funções na esfera \'S POT. 2\'. Cada um destes produtos é definido atravéz de uma correspondência de símbolos, que é uma aplicação linear bijetiva entre operadores lineares num espaço de Hilbert complexo de dimensão n + 1, ou seja matrizes complexas (n + 1) × (n + 1), e polinômios complexos de grau próprio n definidos na 2-esfera, PolyC(\'S POT. 2\')n, satisfazendo algumas propriedades básicas, como equivariância pela ação do grupo de rotações SO(3), preservação das estruturas reais e normalização [12]. Mais geralmente, toda correspondência define um produto associativo mas não comutativo em PolyC(\'S POT. 2\')n induzido do produto de operadores, chamado de produto twisted em PolyC(\'S POT. \'2)n. Cada um destes produtos twisted, por sua vez, pode ser escrito na forma integral f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, onde f, g PolyC(\'S POT. 2)n, \'n IND. 1\',\' n IND. 2\', n \'S POT. 2\'. Em tal representação integral, todas as propriedades do produto twisted são convertidas em propriedades do trikernel integral L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' C. Os produtos twisted estudados nesta tese são os produtos induzidos pela correspondência padrão de Stratonovich e a correspondência padrão de Berezin, respectivamente, que num certo limite assintótico 2j = n definem deformações estritas da álgebra de Poisson de \'S POT. 2\' [12]. Para cada um deste dois produtos, denotados por \'n SOB. 1\' e \'n SOB. b\' respectivamente, seu trikernel integral é denotado por L \'SOB. j 1\' e L \'j SOB. ~b, respectivamente. A primeira parte desta tese consistiu em desenvolver fórmulas mais tratáveis para L \'j SOB. 1\' e L j SOB. b\' nos casos de número de spin j = 1/2, 1, 3/2, 2, fórmulas estas escritas em termos de funções de dois e de três pontos, invariantes por SO(3), como produtos escalares e determinantes. Nossa esperança inicial era de que pudéssemos encontar padrões que nos permitissem inferir fórmulas fechadas para cada um destes trikernels, válidas para qualquer j, ou pelo menos que nos permitissem inferir fórmulas assintóticas para estes trikernels quando 2j = n . Porém, o grau de complexidade das fórmulas desenvolvidas se mostrou fortemente crescente com j, frustrando nossas expectativas iniciais. Partimos então para uma exploração preliminar de um tipo de aproximação assintótica destes produtos de certas funções oscilatórias na esfera. Mais precisamente, na segunda parte desta tese, preparamos e estudamos preliminarmente o produto de Stratonovich e o produto de Berezin (assim como o produto pontual) de dois harmônicos esféricos, \'Y POT. m1 INF. l1\' e \'Y POT. m2 INF. l2 PolyC(\'S POT. 2\')n, no limite assintótico quando tanto \'l IND. 1\' como \'l IND. 2\' tendem a infinito linearmente com n (mantendo \'l IND. i\' n). Este tipo de assintótica para tais produtos, que faz parte do que chamamos mais geralmente de high-l asymptotics, difere muito do tipo de assintótica estudada de forma detalhada em [12], na qual n , mas \'l IND. 1\' e \'l IND. 2\' são mantidos finitos. Então, a partir de um exemplo particular para nossa exploração preliminar, levantamos uma conjectura sobre como estes produtos se comparam no limite assintótico quando \'l IND. 1\' e \'l IND. 2\' tendem para infinito linearmente com o número de spin j / This thesis is about the Stratonovich and the Berezin products of functions on the 2-sphere. Each one of these products is defined via a spin-j symbol correspondence, a linear bijective map from the space of operators on an (n + 1)-dimensional complex Hilbert Space, i.e. (n + 1) × (n + 1) complex matrices, and the space of complex polynomials on \'S POT. 2\' of proper degree n, denoted PolyC(\'S POT. 2\')n, satisfying certain basic properties like equivariance under the action of SO(3), preservation of real structures and normalization [12]. More generally, every spin-j symbol correspondence defines an associative noncommutative product on PolyC(\'S POT. 2\')n induced from the operator product, which is called a twisted product on PolyC(\'S POT. 2\')n. Each twisted product can be written in integral form as f g(n) = \'INT. INF. S POT. 2 X \'S POT. 2\' f(\'n IND. 1\'\') g (\'n IND. 2\'), n) L (\'n IND. 1\', \'n IND.2\', n) \'dn IND.1\' \'dn IND. 2, where f, g PolyC(\'S POT. 2) \' > OR =\' n, \'n IND. 1\',\' n IND. 2\', n \'IT BELONGS\' \'S POT. 2\'. In such and an integral representation, all properties of the twisted product are translated to properties of its integral trikernel L : \'S POT. 2\' × \'S POT. 2\' × \'S POT. 2\' \'ARROW\' C. The twisted products studied in this thesis are the ones obtained via the standard Stratonovich-Weyl and the standard Berezin symbol correspondences, respectively, which in a certain asymptotic limit 2j = n \'ARROW\' \'INFINITY\' define strict deformation quantizations of \'S POT. 2\', i.e. strict deformations of the Poisson algebra of \'S POT. 2\' [12]. For 10 each of these two products on PolyC(\'S POT. 2\') \'< OR =\' n, denoted by * \'n SOB. 1\' and \'n SOB. b\' respectively, its integral trikernel is denoted by L\'j SOB. 1\' and L\'j SOB. b\' , respectively. In the first part of this thesis, we obtained better formulae for these trikernels, for values of spin number j = 1/2, 1/3/1, 2. These formulas are written in terms of SO(3)-invariant functions of two and three points on \'S POT. 2\', like scalar products and determinants. We initially hoped to be able to obtain closed formulae for these trikernels which would be valid for every j, or at least be able to infer asymptotic formulas for these trikernels when 2j = n \'ARROW\' \' INFINITY\' . However, the degree of complexity of the formulae we have obtained increases strongly with j, frustrating our initial expectations. We thus started on a preliminary investigation of a kind of asymptotic approximation for these products of certain oscillatory functions on the sphere. More precisely, in the second part of this thesis, we prepared and preliminarily studied the Stratonovich product and the Berezin product (as well as the pointwise product) of spherical harmonics Y \'m1 SOB. l1\' and Y \'m2 SOB. l2\' PolyC(\'S POT. 2\')n, in the asymptotic limit when l1 and l2 tend to infinity linearly with n (keeping \'l IND. i\' \'< OR =\' n). This kind of asymptotics for these products, belonging to what we more generally callhigh-l asymptotics, differs drastically from the kind of asymptotics studied in detail in [12], in which n \'ARROW\' \'INFINITY\' but \'l IND. 1\' and \'l IND. 2\' are kept finite. Then, based on a particular example of our preliminary exploration, we advanced a conjecture on how these products behave and compare with each other, in the asymptotic limit when \'l IND. \'1\' and \'l IND. 2\' tend to infinity linearly with the spin number j
3

Asymptotic Expansions of Berezin Transforms

Jonathan Arazy, Bent Orsted, jarazy@math.haifa.ac.il 31 July 2000 (has links)
No description available.
4

Weighted Bergman Kernels and Quantization

Miroslav Englis, englis@math.cas.cz 05 September 2000 (has links)
No description available.
5

Matrix Balls, Radial Analysis of Berezin Kernels, and Hypergeometric

Yurii A. Neretin, neretin@main.mccme.rssi.ru 21 December 2000 (has links)
No description available.
6

Algumas Aplicações de Integrais de Trajetória Grassmannianas na Teoria Quântica Moderna / Some Applications of Grassmannianas Trajectory Integrals in Modern Quantum Theory

Paulo Barbosa Barros 29 October 1998 (has links)
Este trabalho é dedicado à aplicação de integrais de trajetória de Grassmann para o cálculo de operadores relevantes aos problemas da teoria quântica relativística. Primeiramente uma visão geral detalhada do método é fornecida. Então concentramos nas definições e aplicações das integrais de trajetória sobre as variáveis de Grassmann. Discutimos, em detalhe, um importante papel das integrais de trajetória de Grassmann na representação de propagadores de partículas relativísticas. Derivamos o chamado fatores de spin para tais representações, fazendo as integrações Grasmannianas. Uma contribuição completamente original foi feita aplicando tais integrais ao cálculo de operadores. Derivamos, desta forma, um conjunto de fórmulas novas para as funções de operadores das matrizes y. A aplicações de tais fórmulas são apresentadas. / This work is devoted to an application of Grassmann path integrals to operator calculus relevant to problems of relativistic quantum theory. A detailed survey of path integral method is given first. Then we concentrate ourselves on definitions and applications of path integrals over Grassmann variables. We discuss in detail an important role of Grassmann path integrals in representations of relativistic particle propagators. We derive the so called spin factors for such representations doing Grassmann integrations. A completely original contribution was made in application of such integrals to operator calculus. We have derived in such a way a set of new formulas for operator functions of y-matrices. Applications of such formulas are presented.
7

Algumas Aplicações de Integrais de Trajetória Grassmannianas na Teoria Quântica Moderna / Some Applications of Grassmannianas Trajectory Integrals in Modern Quantum Theory

Barros, Paulo Barbosa 29 October 1998 (has links)
Este trabalho é dedicado à aplicação de integrais de trajetória de Grassmann para o cálculo de operadores relevantes aos problemas da teoria quântica relativística. Primeiramente uma visão geral detalhada do método é fornecida. Então concentramos nas definições e aplicações das integrais de trajetória sobre as variáveis de Grassmann. Discutimos, em detalhe, um importante papel das integrais de trajetória de Grassmann na representação de propagadores de partículas relativísticas. Derivamos o chamado fatores de spin para tais representações, fazendo as integrações Grasmannianas. Uma contribuição completamente original foi feita aplicando tais integrais ao cálculo de operadores. Derivamos, desta forma, um conjunto de fórmulas novas para as funções de operadores das matrizes y. A aplicações de tais fórmulas são apresentadas. / This work is devoted to an application of Grassmann path integrals to operator calculus relevant to problems of relativistic quantum theory. A detailed survey of path integral method is given first. Then we concentrate ourselves on definitions and applications of path integrals over Grassmann variables. We discuss in detail an important role of Grassmann path integrals in representations of relativistic particle propagators. We derive the so called spin factors for such representations doing Grassmann integrations. A completely original contribution was made in application of such integrals to operator calculus. We have derived in such a way a set of new formulas for operator functions of y-matrices. Applications of such formulas are presented.
8

Low-energy spectrum of Toeplitz operators / Le spectre à basse énergie des opérateurs de Toeplitz

Deleporte-Dumont, Alix 29 March 2019 (has links)
Les opérateurs de Berezin--Toeplitz permettent de quantifier des fonctions, ou des symboles, sur des variétés kähleriennes compactes, et sont définies à partir du noyau de Bergman (ou de Szeg\H{o}). Nous étudions le spectre des opérateurs de Toeplitz dans un régime asymptotique qui correspond à une limite semiclassique. Cette étude est motivée par le comportement magnétique atypique observé dans certains cristaux à basse température. Nous étudions la concentration des fonctions propres des opérateurs de Toeplitz, dans des cas où les effets sous-principaux (du même ordre que le paramètre semiclassique) permet de différencier entre plusieurs configurations classiques, un effet connu en physique sous le nom de sélection quantique Nous exhibons un critère général pour la sélection quantique et nous donnons des développements asymptotiques précis de fonctions propres dans le cas Morse et Morse--Bott, ainsi que dans un cas dégénéré. Nous développons également un nouveau cadre pour le traitement du noyau de Bergman et des opérateurs de Toeplitz en régularité analytique. Nous démontrons que le noyau de Bergman admet un développement asymptotique, avec erreur exponentiellement petite, sur des variétés analytiques réelles. Nous obtenons aussi une précision exponentiellement fine dans les compositions et le spectre d'opérateurs à symbole analytique, et la décroissance exponentielle des fonctions propres. / Berezin-Toeplitz operators allow to quantize functions, or symbols, on compact Kähler manifolds, and are defined using the Bergman (or Szeg\H{o}) kernel. We study the spectrum of Toeplitz operators in an asymptotic regime which corresponds to a semiclassical limit. This study is motivated by the atypic magnetic behaviour observed in certain crystals at low temperature. We study the concentration of eigenfunctions of Toeplitz operators in cases where subprincipal effects (of same order as the semiclassical parameter) discriminate between different classical configurations, an effect known in physics as quantum selection . We show a general criterion for quantum selection and we give detailed eigenfunction expansions in the Morse and Morse-Bott case, as well as in a degenerate case. We also develop a new framework in order to treat Bergman kernels and Toeplitz operators with real-analytic regularity. We prove that the Bergman kernel admits an expansion with exponentially small error on real-analytic manifolds. We also obtain exponential accuracy in compositions and spectra of operators with analytic symbols, as well as exponential decay of eigenfunctions.
9

Berezin--Toeplitz quantization and noncommutative geometry

Falk, Kevin 11 September 2015 (has links)
Cette thèse montre en quoi la quantification de Berezin--Toeplitz peut être incorporée dans le cadre de la géométrie non commutative.Tout d'abord, nous présentons les principales notions abordées : les opérateurs de Toeplitz (classiques et généralisés), les quantifications géométrique et par déformation, ainsi que quelques outils de la géométrie non commutative.La première étape de ces travaux a été de construire des triplets spectraux (A,H,D) utilisant des algèbres d'opérateurs de Toeplitz sur les espaces de Hardy et Bergman pondérés relatifs à des ouverts Omega de Cn à bord régulier et strictement pseudoconvexes, ainsi que sur l'espace de Fock sur Cn. Nous montrons que les espaces non commutatifs induits sont réguliers et possèdent la même dimension que le domaine complexe sous-jacent. Différents opérateurs D sont aussi présentés. Le premier est l'opérateur de Dirac usuel sur L2(Rn) ramené sur le domaine par transport unitaire, d'autres sont formés à partir de l'opérateur d'extension harmonique de Poisson ou de la dérivée normale complexe sur le bord de Omega.Dans un deuxième temps, nous présentons un triplet spectral naturel de dimension n+1 construit à partir du produit star de la quantification de Berezin--Toeplitz. Les éléments de l'algèbre correspondent à des suites d'opérateurs de Toeplitz dont chacun des termes agit sur un espace de Bergman pondéré. Plus généralement, nous posons des conditions pour lesquelles une somme infinie de triplets spectraux forme de nouveau un triplet spectral, et nous en donnons un exemple. / The results of this thesis show links between the Berezin--Toeplitz quantization and noncommutative geometry.We first give an overview of the three different domains we handle: the theory of Toeplitz operators (classical and generalized), the geometric and deformation quantizations and the principal tools we use in noncommutative geometry.The first step of the study consists in giving examples of spectral triples (A,H,D) involving algebras of Toeplitz operators acting on the Hardy and weighted Bergman spaces over a smoothly bounded strictly pseudoconvex domain Omega of Cn, and also on the Fock space over Cn. It is shown that resulting noncommutative spaces are regular and of the same dimension as the complex domain. We also give and compare different classes of operator D, first by transporting the usual Dirac operator on L2(Rn) via unitaries, and then by considering the Poisson extension operator or the complex normal derivative on the boundary.Secondly, we show how the Berezin--Toeplitz star product over Omega naturally induces a spectral triple of dimension n+1 whose construction involves sequences of Toeplitz operators over weighted Bergman spaces. This result led us to study more generally to what extent a family of spectral triples can be integrated to form another spectral triple. We also provide an example of such triple.
10

Propriétés spectrales des opérateurs de Toeplitz

Barusseau, Benoit 20 May 2010 (has links)
La première partie de la thèse réunit des résultats classiques sur l’espace de Hardy, les espaces modèles et l’espace de Bergman. Puis sur cette base, nous exposons des travaux relatifs aux opérateurs de Toeplitz, en particulier, nous présentons la description du spectre et du spectre essentiel de ces opérateurs sur l’espace de Hardy et de Bergman. La première partie de notre recherche tire son inspiration de deux faits établis pour un opérateur de Toeplitz T. Premièrement, sur l’espace de Hardy, la norme de T, la norme essentielle de T et la norme infinie du symbole de T sont égales. Nous étudions ce cas d’égalité sur l’espace de Bergman pour les opérateurs de Toeplitz à symbole quasihomogène et radial. Deuxièmement, sur l’espace de hardy, le spectre et le spectre essentiel sont fortement liés à l’image du symbole de T. Nous étudions le cas d’égalité entre le spectre et l’image essentielle du symbole pour les symboles quasihomogènes et radials. Pour répondre à ces deux questions, nous utilisons la transformée de Berezin, les coefficients de Mellin et la moyenne du symbole. La dernière partie de la thèse s’interesse au théorème de Szegö qui donne un lien entre les valeurs propres d’une suite de matrices de Toeplitz de taille n, et le symbole de cette suite de matrice. Nous donnons un résultat du même type sur l’espace de Bergman pour les symboles harmoniques sur le disque et continus sur le cercle. Enfin, nous étudions une généralisation de ce théorème en compressant l’opérateur de Toeplitz sur une suite d’espaces modèles de dimension finie. / This thesis deals with the spectral properties of the Toeplitz operators in relation to their associated symbol. In the first part, we give some classical results about Hardy space, model spaces and Bergman space. Afterwards, we expose some results about Toeplitz operator on the Hardy space. In particular, we discuss their spectrum and essential spectrum. Our work is inspired from two facts which have been proved on the Hardy space. First, considering a Toeplitz operator T, the norm, essential norm, spectral radius of T and the supremum of its symbol are equal. Secondly, on the Hardy space, spectrum, essential spectrum and essential range are strongly related. We answer the question of the equality between the norms, the spectral radius and the supremum of the symbol and between spectrum and essential range on the Bergman space. We look at these two properties on the Bergman space when the symbol is radial or quasihomogeneous. We answer these questions using the Berezin transform, the Mellin coefficients and the mean value of the symbol. The last part deals with the classical Szegö theorem which underline a link between the eigenvalues of a Toeplitz matrix sequence and its symbol. We give a result of the same type on Bergman space considering harmonic symbol wich have a continuous extension. We give a generalization, considering the sequence of the compressions of a Toeplitz operator on a sequence of model spaces.

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