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Yousef, Abdelrahman F. "Two problems in the theory of Toeplitz operators on the Bergman space /." Connect to full text in OhioLINK ETD Center, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1242219617.
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Arroussi, Hicham. "Function and Operator Theory on Large Bergman spaces." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/395175.
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The theory of Bergman spaces has been a central subject of study in complex analysis during the past decades. The book [7] by S. Bergman contains the first systematic treat-ment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on a domain. His approach was based on a reproducing kernel that became known as the Bergman kernel function. When attention was later directed to the spaces AP over the unit disk, it was natural to call them Bergman spaces. As counterparts of Hardy spaces, they presented analogous problems. However, although many problems in Hardy spaces were well understood by the 1970s, their counterparts for Bergman spaces were generally viewed as intractable, and only some isolated progress was done. The 1980s saw the emerging of operator theoretic studies related to Bergman spaces with important contributions by several authors. Their achievements on Bergman spaces with standard weights are presented in Zhu's book [77]. The main breakthroughs came in the 1990s, where in a flurry of important advances, problems previously considered intractable began to be solved. First came Hedenmalm's construction of canonical divisors [26], then Seip's description [59] of sampling and interpolating sequences on Bergman spaces, and later on, the study of Aleman, Richter and Sundberg [1] on the invariant subspaces of A2, among others. This attracted other workers to the field and inspired a period of intense research on Bergman spaces and related topics. Nowadays there are rich theories on Bergman spaces that can be found on the textbooks [27] and [22].
Meanwhile, also in the nineties, some isolated problems on Bergman spaces with ex-ponential type weights began to be studied. These spaces are large in the sense that they contain all the Bergman spaces with standard weights, and their study presented new dif-ficulties, as the techniques and ideas that led to success when working on the analogous problems for standard Bergman spaces, failed to work on that context. It is the main goal of this work to do a deep study of the function theoretic properties of such spaces, as well as of some operators acting on them. It turns out that large Bergman spaces are close in spirit to Fock spaces [79], and many times mixing classical techniques from both Bergman and Fock spaces in an appropriate way, can led to some success when studying large Bergman spaces.
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Randriamahaleo, Fanilo rajaofetra. "Opérateurs de Toeplitz sur l'espace de Bergman harmonique et opérateurs de Teoplitz tronqués de rang fini." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0108/document.
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Dans la première partie de la thèse, nous donnons les résultats classiques concernant l’espace de Hardy, les espaces modèles et les espaces de Bergman analytique et harmonique. Les notions de base telles que les projections et les noyaux reproduisant y sont introduites. Nous exposons ensuite nos résultats concernant d’une part, la stabilité du produit et la commutativité de deux opérateurs de Toeplitz quasihomogènes et d’autre part, la description matricielle des opérateurs de Toeplitz tronqués du type "a" "dans le cas de la dimension finieIn the first part of the thesis,we give some classical results concerning theHardy space, models spaces and analytic and harmonic Bergman spaces. The basic concepts such as projections and reproducing kernels are introduced. We then describe our results on the the stability of the product and the commutativity of two quasihomogeneous Toeplitz operators on the harmonic Bergman space. Finally, we give the matrix description of truncated Toeplitz operators of type "a" in the finite dimensional case
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Yousef, Abdelrahman Fawzi. "Two Problems in the Theory of Toeplitz Operators on the Bergman Space." University of Toledo / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1242219617.
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Subedi, Krishna Subedi. "Hyponormality and Positivity of Toeplitz operators via the Berezin transform." University of Toledo / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1532963068992661.
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Le, Van An. "Petits espaces de Fock, petits espaces de Bergman et leurs opérateurs." Thesis, Aix-Marseille, 2019. http://theses.univ-amu.fr.lama.univ-amu.fr/191210_LE_604try554eejyoj865ovdfq987fxy_TH.pdf.
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Nous étudions les mesures de Carleson et les opérateurs de Toeplitz sur la classe des espaces de Bergman dite de petite taille, introduits récemment par Seip. On obtient une caractérisation des mesures de Carleson qui étend les résultats de Seip à partir du disque unité de mathbb C à la boule unité mathbb Bn de mathbb Cn. Nous utilisons cette caractérisation pour donner les conditions nécessaires et suffisantes à la continuité et à la compacité des opérateurs de Toeplitz. Enfin, nous étudions l’appartenance des opérateurs Toeplitz aux classes de Schatten d'ordre p pour 1We study the Carleson measures and the Toeplitz operators on the class of the so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disk of mathbb C to the unit ball mathbb Bn of mathbb Cn. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten p classes membership of Toeplitz operators for 1
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Jones, Matthew Michael. "Composition operators on weighted Bergman spaces." Thesis, University College London (University of London), 1999. http://discovery.ucl.ac.uk/1363351/.
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In the late 1960’s, E.A. Nordgren and J.V. Ryff studied composition operators on the Hardy space H2. They provided upper and lower bounds on the norms of general composition operators and gave the exact norm in the case where the symbol map is an inner function. Composition operators themselves, on various other spaces, have been studied by many authors since and much deep work has been done concerning them. Recently, however B.D. MacCluer and T. Kriete have developed the study of composition operators on very general weighted Bergman spaces of the unit disk in the complex plane. My starting point is this work. Composition operators serve well to link the two areas of analysis, operator theory and complex function theory. The products of this link lie deep in complex analysis and are diverse indeed. These include a thorough study of the Schr¨oeder functional equation and its solutions, see [16] and the references therein, in fact some of the well known conjectures can be linked to composition operators. Nordgren, [12], has shown that the Invariant Subspace Problem can be solved by classifying the minimal invariant subspaces of a certain composition operator on H2, and de Branges used composition operators to prove the Bieberbach conjecture. In this thesis, I use various methods from complex function theory to prove results concerning composition operators on weighted Bergman spaces of the unit disk, the main result is the confirmation of two conjectures of T. Kriete, which appeared in [7]. I also construct, in the final chapter, inner functions which map one arbitrary weighted Bergman space into another.
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Oliver, Vendrell Roc. "Hankel operators on vector-valued Bergman spaces." Doctoral thesis, Universitat de Barcelona, 2017. http://hdl.handle.net/10803/471520.
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The main goal of this work is to study vector-valued Bergman spaces and to obtain the weak factorization of these spaces. In order to do that we need to study small Hankel operators with operator-valued holomorphic symbols. We also study the big Hankel operator acting on vector-valued Bergman spaces.
In Chapter 1 we collect all the previous results and notations needed to follow the rest of the manuscript. More concretely, some of the topics covered in this chapter are the Bochner integral, the integral for vector-valued functions appearing first in Bochner; the Bergman metric, results of the metric used in Bn; harmonic and subharmonic function; basic notions of differentiation, where the differential operators R(a, t) are presented which is important in the next chapters and in the final section we recall some topics on Banach spaces, as the Rademacher type and cotype of a Banach space and some other related results.
Having all that in mind, in Chapter 2, the vector-valued Bergman spaces are presented. The vector-valued Bloch type spaces play a similar role and therefore we dedícate one full chapter to these spaces. Chapter 3 is devoted to present and characterize the vector-valued Bloch type spaces. Since we mention Hankel operators, in Chapter 4 we prove the characterization of the boundedness of the small Hankel operator with analytic operator-valued symbols between vector-valued Bergman spaces (of different type). We explain what
this means in the following.
Another very important consequence of the boundedness of the small Hankel operator between vector-valued Bergman spaces is shown in Chapter 5. We establish the weak factorization of the vector-valued Bergman spaces. Factorization of analytic functions is a very big topic and many people worked on it during many years and it is known to have many applications.
Therefore, in Chapter 6 we fully characterize the boundedness of the big Hankel operator on vector-valued Bergman spaces in terms of its operator-valued holomorphic symbol for all cases of p > 1 and q > 1, and so we solve and generalize the previous problem.
Finally, in Chapter 7 we discuss some open problems we have not been able to solve, as well as some other interesting problems in the same line as this work in order to look on the future.
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Kraemer, Daniel [Verfasser], and Jörg [Akademischer Betreuer] Eschmeier. "Toeplitz operators on Hardy spaces / Daniel Kraemer ; Betreuer: Jörg Eschmeier." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2020. http://d-nb.info/1209947455/34.
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Tattersall, Joshua Malcolm. "Toeplitz and Hankel operators on Hardy spaces of complex domains." Thesis, University of Leeds, 2015. http://etheses.whiterose.ac.uk/11498/.
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The major focus is on the Hardy spaces of the annulus {z : s < |z| < 1}, with the measure on the boundary being Lebesgue measure normalised such that each boundary has weight 1. There is also consideration of higher order annuli, the Bergmann spaces and slit domains. The focus was on considering analogues of classical problems in the disc in multiply connected regions. Firstly, a few factorisation results are established that will assist in later chapters. The Douglas-Rudin type factorisation is an analogue of factorisation in the disc, and the factorisation of H1 into H2 functions are analogues of factorisation in the disc, whereas the multiplicative factorisation is specific to multiply connected domains. The Douglas-Rudin type factorisation is a classical result for the Hardy space of the disc, here it is shown for the domain {z : s < |z| < 1}. A previous factorisation for H1 into H2 functions exists in [4], an improved constant not depending on s is found here. We proceed to investigate real-valued Toeplitz operators in the annulus, focusing on eigenvalues and eigenfunctions, including for higher order annuli, and amongst other results the general form of an eigenfunction is determined. A paper of Broschinski [10] details the same approach for the annulus {z : s < |z| < 1} as here, but does not consider higher genus settings. There exists work such as in [6] and [5] detailing an alternative approach to eigenvalues in a general setting, using theta-functions, and does not detail the eigenfunctions. After this, kernels of a more general symbol are considered, compared to the disc, and Dyakanov’s theorem from the disc is extended for the annulus. Hankel operators are also considered, in particular with regards to optimal symbols. Finally, analogues of results from previous chapters are considered in the Bergman space, and the Hardy space of a slit annulus.
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Fedchenko, Dmitry, and Nikolai Tarkhanov. "A Class of Toeplitz Operators in Several Variables." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6893/.
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We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the
index theory.
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Fulsche, Robert [Verfasser]. "Toeplitz operators and generated algebras on non-Hilbertian spaces / Robert Fulsche." Hannover : Gottfried Wilhelm Leibniz Universität, 2020. http://d-nb.info/1223090264/34.
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Deleporte-Dumont, Alix. "Low-energy spectrum of Toeplitz operators." Thesis, Strasbourg, 2019. http://www.theses.fr/2019STRAD004/document.
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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 propresBerezin-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
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Harutyunyan, Anahit V. "Toeplitz operators and division theorems in anisotropic spaces of holomorphic functions in the polydisc." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2611/.
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This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.
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Barusseau, Benoit. "Propriétés spectrales des opérateurs de Toeplitz." Thesis, Bordeaux 1, 2010. http://www.theses.fr/2010BOR14027/document.
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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 finieThis 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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Liang, Xiaoming. "A class of weighted Bergman spaces, reducing subspaces for multiple weighted shifts, and dilatable operators." Diss., Virginia Tech, 1996. http://hdl.handle.net/10919/39164.
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This thesis consists of four chapters. Chapter 1 contains the preliminaries. We give the background, notation and some results needed for this work, and we describe our main results of this thesis.
In Chapter 2 we will introduce a class of weighted Bergman spaces. We then will discuss some properties about the multiplication operator, Mz , on them. We also characterize the dual spaces of these weighted Bergman spaces.
In Chapter 3 we will characterize the reducing subspaces of multiple weighted shifts. The reducing subspaces of the Bergman and the Dirichlet shift of multiplicity N are portrayed from this characterization.
In Chapter 4 we will introduce the class of super-isometrically dilatable operators and describe their elementary properties. We then will discuss an equivalent description of the invariant subspace lattice for the Bergman shift. We will also discuss the interpolating sequences on the bidisk. Finally, we will examine a special class of super-isometrically dilatable operators. One corollary of this work is that we will prove that the compression of the Bergman shift on two compliments of two invariant subspaces are unitarily equivalent if and only if the two invariant subspaces are equal.Ph. D.
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Bauer, Wolfram [Verfasser]. "Toeplitz Operators on finite and infinite dimensional spaces with associated Psi*-Fréchet Algebras / Wolfram Bauer." Aachen : Shaker, 2006. http://d-nb.info/1186587393/34.
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Tytgat, Romaric. "Trace de Dixmier d'opérateurs de Hankel." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4772/document.
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Nous nous intéressons aux opérateurs de Hankel $H_{bar{f}}$ de symbole anti holomorphe $bar{f}$ et regardons l'espace de Dixmier $mathcal{D}^{p}$ associé ($pgeq1$), c'est à dire l'ensemble des $f$ tel que $|H_{bar{f}}|^{p}$ soit dans l'idéal de Macaev $mathcal{S}^{+}_{1}$. Notre approche est de voir l'espace de Dixmier comme une certaine limite des classes de Schatten. Quand $f in mathcal{D}^{p}$, nous étudions $Tr_{omega}(|$H_{bar{f}}$|^{p})$ la trace de Dixmier de $|H_{bar{f}}|^{p}$. Nous redémontrons certains résultats classiques quand $f$ est holomorphe sur le disque alors que nous donnons de nouveaux résultats quand $f$ est entière. Nous utilisons notre méthode pour étudier l'espace de Dixmier du petit opérateur de Hankel, des opérateurs de Toeplitz $T_{varphi}$ ($varphi$ définie sur le disque ou sur le plan complexe tout entier) ainsi que pour l'opérateur de compositionWe study Hankel operators $H_{bar{f}}$ with anti holomorphic symbol $bar{f}$ and we are interested to the Dixmier space $mathcal{D}^{p}$ ($pgeq1$), the set of functions $f$ such that $|H_{bar{f}}|^{p} in mathcal{S}^{+}_{1}$ the Macaev ideal. We look Dixmier space as a limit of Schatten class. When $f in mathcal{D}^{p}$, we study $Tr_{omega}(|$H_{bar{f}}$|^{p})$ the Dixmier trace of $|H_{bar{f}}|^{p}$. We have different results when $f$ is an entire or a holomorphic function of the unit disk in the complex plan. We study also the Dixmier space of the little Hankel operator, Toeplitz operator and composition operator
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Casseli, Irène. "Eléments sur la transformée de Berezin et sur les opérateurs de Toeplitz dans des espaces de fonctions polyanalytiques." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0578.
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Les fonctions polyanalytiques entières généralisent les fonctions entières dans la mesure où elles sont les solutions sur le plan complexe \mathbb{C} de l'équation de Cauchy-Riemann à l'ordre n, de la forme { partial} nf / \partial \overline{z} n = 0. Un espace de Fock polyanalytique F2 {\alpha,n} est, par analogie avec le cas classique, le sous-espace fermé de l'espace de Hilbert L^2 (\mathbb{C},d\mu \alpha), où \mu \alpha est une mesure de probabilité gaussienne sur \mathbb{C} de paramètre alpha>0, formé des fonctions polyanalytiques entières d'ordre n. L'objet de cette thèses est l'étude d'éléments classiques de la théorie des opérateurs tels que la transformée de Berezin et les opérateurs de Toeplitz dans le cadre particulier des espaces de Fock polyanalytiques. Dans ce manuscrit, il est montré en particulier que les points fixes de la transformée de Berezin qui appartiennent aux espaces de Lebesgue sont les fonctions nulles ou éventuellement constantes. Concernant les opérateurs de Toeplitz, le problème de Sarason est étudié. Etant donné une fonction f, l'opérateur de Toeplitz de symbole f est formellement défini par T {alpha,n} f(h)=P {alpha,n}(f h), où P {alpha,n} est la projection orthogonale de L^2(\mathbb{C},d\mu {alpha}) sur F^2 {alpha,n}. Le problème de Sarason consiste à donner une condition nécessaire et suffisante sur f et g pour que le produit d'opérateurs de symboles f et bar g soit continuEntire polyanalytic functions generalize entire functions in that they are solutions of "Cauchy-Riemann equations of order n, of the form {\partial}^n f / \partial \overline{z}^n = 0, over the whole complex plane \mathbb{C}. Polyanalytic Fock space F^2_{\alpha,n} is, by analogy with the classical case, the closed subspace of the Hilbert space L^2(\mathbb{C},d\mu_\alpha), where \mu_\alpha is a Gaussian probability measure over \mathbb{C} with weight \alpha>0, of polyentire functions of order n. The aim of this PhD thesis is the study of classical objects of operator theory such that the Berezin transform and Toeplitz operators in the particular case of polyanalytic Fock spaces. In this written, it is shown among other results, that the L^p fixed points of the Berezin transform are constant functions. Concerning Toeplitz operators, the Sarason problem is studied. Given a function f, the Toeplitz operator with symbol f is formally defined by T^n_f(h)=P_{F^2_n}(f h), where P_{F^2_n} is the orthogonal projection from L^2(\mathbb{C},d\mu) on to F^2_n. The so-called Sarason's problem consists in finding necessary and sufficient conditions on the symbols f and g for the Toeplitz product with symbols f and \bar g to be bounded in the Fock space
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Le, Trieu Long. "Toeplitz operators on the Hardy and Bergman spaces." 2007. http://proquest.umi.com/pqdweb?did=1335360441&sid=17&Fmt=2&clientId=39334&RQT=309&VName=PQD.
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Thesis (Ph.D.)--State University of New York at Buffalo, 2007.Title from PDF title page (viewed on Nov. 14, 2007) Available through UMI ProQuest Digital Dissertations. Thesis adviser: Xia, Jingbo. Includes bibliographical references.
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Chailuek, Kamthorn. "An extension of Bergman spaces and their Toeplitz operators." 2007. http://etd.nd.edu/ETD-db/theses/available/etd-07192007-033805/.
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Issa, Hassan. "The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions." Doctoral thesis, 2012. http://hdl.handle.net/11858/00-1735-0000-000D-F066-5.
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Bauer, Wolfram [Verfasser]. "Toeplitz operators on finite and infinite dimensional spaces with associated Ψ*-Fréchet [Psi*-Fréchet] algebras / Wolfram Bauer." 2006. http://nbn-resolving.de/urn:nbn:de:hebis:77-10375.
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