Submitted by Marco Antônio de Ramos Chagas (mchagas@ufv.br) on 2018-06-25T17:05:26Z No. of bitstreams: 1 texto completo.pdf: 1015662 bytes, checksum: 7494a65a5f15ab8325f4a81613dec9c3 (MD5) / Made available in DSpace on 2018-06-25T17:05:26Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1015662 bytes, checksum: 7494a65a5f15ab8325f4a81613dec9c3 (MD5) Previous issue date: 2018-03-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho estuda-se ferramentas sobre a importância do espaço Hardy- Hilbert e da análise funcional para abordagem do espectro do operador Toeplitz. Durante o estudo, é explícito que o espectro de tal operador depende de seu símbolo, estudando o espectro para o operador Toeplitz analítico(símbolo analítico), coanalítico(conjugado de seu símbolo analítico), autoadjunto(símbolo real) e operador Toeplitz com símbolo contínuo. Em seguida, introduz-se definições do operador simétrico complexo, para logo responder perguntas como: Quando um operador Toeplitz é simétrico complexo sobre o espaço de Hilbert H 2 ?, quando um operador Toeplitz simétrico complexo é normal? / In this work we study the importance of the Hardy-Hilbert space and the functional analysis to approach the spectrum of the Toeplitz operator, during the study, it is explicit that the spectrum of such operators depend on its symbol, studying the spectrum for the analytical Toeplitz operator (analytical symbol), coanalytic (conjugate of its analytic symbol), autoadjunto (real symbol) and Toeplitz operator with continuous symbol. Next, we introduce the definition of a complex symmetric operator and then answer questions such as: When a Toeplitz operator is complex symmetric on the Hilbert space H 2 ?, when a complex symmetric Toeplitz operator is normal?

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / El objetivo de esta tesis es presentar condiciones para analizar la existencia de valores propios en algunas familias de sistemas de Bratteli-Vershik minimales. Ésto expande los resultados obtenidos hasta el día de hoy en ciertas familias de sistemas de Bratteli-Vershik minimales de rango finito, como son los sistemas estacionarios, linealmente recurrentes y Toeplitz. En los primeros dos capítulos de la tesis se presentan estos resultados preliminares, junto con construcciones relevantes, varias de ellas mencionadas en la literatura, pero nunca escritas formalmente. Al hacer esto aparecen un par de ejemplos y métodos particulares, que se adjuntan pretendiendo aportar un poco más al entendimiento de los sistemas de Bratteli-Vershik. En el tercer capítulo se presenta una caracterización de la ocurrencia de un valor propio en un sistema de Bratteli-Vershik minimal de rango finito general, sin distinguir entre valores propios continuos y no-continuos. Esta caracterización tiene la ventaja de estar expresada en términos de elementos combinatoriales relacionados naturalmente a un diagrama de Bratteli, como son sus matrices de incidencia, su orden, y las alturas de las torres de Kakutani-Rokhlin asociadas. En el último capítulo se analizan los valores propios de los sistemas de Toeplitz minimales de rango finito. A partir del ajuste natural de la condición general para sistemas de rango finito, y del análisis de la subfamilia de diagramas esencialmente cíclicos, se establece una caracterización de la ocurrencia de valores propios no-continuos en los sistemas Toeplitz minimales de rango finito, desde diferentes puntos de vista. Por ejemplo, se establece que los únicos sistemas de Toeplitz que poseen valores propios no-continuos son, salvo conjugación, los que provienen de diagramas esencialmente cíclicos. Finalmente se establece una relación entre los valores propios no-continuos de un sistema Toeplitz minimal de rango finito, y la cantidad de medidas ergódicas que dicho sistema posee.

We proved that the topological conjugacy relation on $T_1$, a subclass of Toeplitz subshifts, is hyperfinite, extending Kaya's result that the topological conjugate relation of Toeplitz subshifts with growing blocks is hyperfinite. A close concept about the topological conjugacy is the flip conjugacy, which has been broadly studied in terms of the topological full groups. Particularly, we provided an equivalent characterization on Toeplitz subshifts with single hole structure to be flip invariant.

Eigenstructure based techniques have been studied extensively in the last decade to estimate the number and locations of incoming radiating sources using a passive sensor array. One of the early limitations was the computational load involved in arriving at the eigendecompositions. The introduction of VLSI circuits and parallel processors however, has reduced the cost of computation A tremendously. As a consequence, we study eigendecomposition algorithms with highly parallel and A localized data flow, in order to take advantage of VLSI capabilities. This dissertation presents a fast Recursive/Iterative Toeplitz (Hermitian) Eigenspace (RITE) algorithm, and its extension to the generalized strongly regular eigendecomposition situation (C-RITE). Both procedures exhibit highly parallel structures, and their applicability to fast passive array processing is emphasized. The algorithms compute recursively in increasing order, the complete (generalized) eigendecompositions of the successive subproblems contained in the maximum size one. At each order, a number of independent, structurally identical, non-linear problems is solved in parallel. The (generalized) eigenvalues are found by quadratically convergent iterative search techniques. Two different search methods, a restricted Newton approach and a rational approximation based technique are considered. The eigenvectors are found by solving Toeplitz systems efficiently. The multiple minimum (generalized) eigenvalue case and the case of a cluster of small (generalized) eigenvalues are treated also. Eigenpair residual norms and orthonormality norms in comparison with IMSL library routines, indicate good performance and stability behavior for increasing dimensions for both the RITE and C-RITE algorithms. Application of the procedures to the Direction Of Arrival (DOA) identification problem, using the MUSIC algorithm, is presented. The order-recursive properties of RITE and C-RITE permit estimation of angles for all intermediate orders imbedded in the original problem, facilitating the earliest possible estimation of the number and location of radiating sources. The detection algorithm based on RITE or C-RITE can then stop, thereby minimizing the overall computational load to that corresponding to the smallest order for which angle of arrival estimation is indicated to be reliable. Some extensions of the RITE procedure to Hermitian (non-Toeplitz) matrices are presented. This corresponds in the array processing context to correlation matrices estimated from non-linear arrays or incoming signals with non-stationary characteristics. A first—order perturbation approach and two Subspace Iteration (SI) methods are investigated. The RITE decomposition of the Toeplitzsized (diagonally averaged) matrix is used as a starting point. Results show that the SI based techniques lead to good approximation of the eigen-information, with the rate of convergence depending upon the SNR ar1d the angle difference between incoming sources, the convergence being faster than starting the SI method from an arbitrary initial matrix. / Ph. D.

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.

This dissertation is concerned with invertibility and one-sided invertibility of Toeplitz operators over the half plane whose generating functions admit homogenous discontinuities, and with stability of their pseudo finite sections. The invertibility criterium is given in terms of invertibility of a family of one dimensional Toeplitz operators with piecewise continuous generating functions. The one-sided invertibility criterium is given it terms of constraints on the partial indices of certain Toeplitz operator valued function.

A new way of proof of Szegö-type theorems is presented. The idea of the proof is based on the construction of "almost" inverse operator to the finite section T_n(a) of a Toeplitz operator T(a), which is close to the inverse operator in the trace norm (these "almost" inverses are well-known). This way of proof gives the possibility to write another representation for the second constant E_f(a), and in the scalar case to receive a shorter representation. Another observation is that the convergence in these theorems is strongly dependent on the smoothness of the generating function a.

Witten, Reshetikhin et Turaev ont défini des invariants des variétés topologiques de dimension 3, dits "quantiques" qui s'étendent en une structure de TQFT, c'est-à-dire un foncteur monoïdal d'une catégorie de cobordismes vers la catégorie des espaces vectoriels complexes. Nous étudions ici leur asymptotique. Dans ce cadre, les courbes sur une surface induisent des endomorphismes des espaces de TQFT, appelés opérateurs courbes, qui sont l'un des objets centraux du mémoire. Tous ces invariants dépendant d'un paramètre entier r, on s'intéresse à leur comportement quand r tend vers l'infini. On s'aperçoit alors que les invariants quantiques sont liés à des objets plus géométriques, comme les espaces des modules des représentations dans SU2 du groupe fondamental d'une surface. La première partie de la thèse introduit la notion de TQFT et les invariants de Witten-Reshetikhin-Turaev, puis donne des rudiments de géométrie de l'espace des modules SU2 d'une surface et de quantification géométrique. La deuxième partie présente un résultat sur l'asymptotique des coefficients de matrices des opérateurs courbes en TQFT. A partir de calcul d'écheveau et d'un théorème de Bullock, on relie les deux premiers termes de leur développement aux fonctions traces associées aux multicourbes. Cette thèse aboutit dans la troisième partie à un résultat asymptotique pour les coefficients de matrices des représentations quantiques. Un modèle géométrique est proposé pour les espaces de TQFT associés aux surfaces, et il est montré que les opérateurs courbes s'identifient alors à des opérateurs de Toeplitz. Des méthodes standards d'analyse semi-classiques permettent d'en déduire le résultat. / In this thesis we study the asymptotics of some invariants of 3-manifolds, known as "quantum invariants" which were defined by Witten, Reshetikhin and Turaev. These invariants are part of a TQFT structure, that is a monoidal functor for a category of cobordism to the category of complex vector spaces. In this setting, curves on surfaces induce endomorphisms of TQFT vector spaces, called curve operators, which are one of the main object in our study. All these invariants depend of an integer parameter r, and we are interested in their behavior when r tends to infinity. We can then see that quantum invariants are related to more geometric objects, like the moduli space of conjugacy classes of SU2 representations of the fundamental group of a surface. The thesis is divided in 3 parts: in the first one we introduce the notion of TQFT and the Witten-Reshetikhin-Turaev invariants, then we give basic properties of the SU2-moduli spaces and explain the general approach of geometric quantification. In the second one we present a result on the asymptotics of matrix coefficients of curve operators. Using skein calculus and a theorem of Bullock, we express the first two terms of their expansion in terms of trace functions on the SU2-moduli space associated to multicurves. The final part gives an asymptotic expansion of matrix coefficents of quantum representations. A geometric model for TQFT vector spaces is defined, and we show that curve operators can be seen as Toeplitz operators in this model. Standard tools of semi-classical analysis allow us to deduce the result from this.

This paper is devoted to asymptotic estimates for the condition numbers $\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$ of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where $\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ , or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues, and the finite section method are discussed. We also consider Wiener-Hopf integral operators and multidimensional Toeplitz operators.

On s'intéresse à la théorie spectrale d'opérateurs semi-classiques non auto-adjoints en dimension un et plus précisément aux développements asymptotiques des valeurs propres. Ces derniers font intervenir des objets géométriques issus de la mécanique classique dans l'espace des phases complexifié et correspondent à une généralisation des conditions de quantification de Bohr-Sommerfeld au cadre non auto-adjoint. Plus précisément, dans un premier temps, on étudie le spectre de perturbations non auto-adjointes d'opérateurs pseudo-différentiels auto-adjoints en dimension un à l'aide de techniques d'analyse microlocale analytique et en corollaire, on établit que pour des perturbations PT-symétriques d'opérateurs auto-adjoints, le spectre est réel. Ensuite, on présente des conditions de quantification de Bohr-Sommerfeld pour des perturbations non auto-adjointes d'opérateurs de Berezin-Toeplitz du plan complexe auto-adjoints. Dans un second temps, on s'intéresse aux différentes quantifications du tore et plus précisément à la quantification de Berezin-Toeplitz du tore, à la quantification de Weyl classique du tore et à la quantification de Weyl complexe du tore. On établit des liens entre ces différentes quantifications notamment grâce à la transformée de Bargmann, puis à l'aide de simulations numériques, on met en évidence une conjecture sur des conditions de quantification de Bohr-Sommerfeld pour des perturbations non auto-adjointes d'opérateurs de Berezin-Toeplitz du tore auto-adjoints. / We interest ourselves in the spectral theory of non self-adjoint semi-classical operators in dimension one and in asymptotic expansions of eigenvalues. These expansions are written in terms of geometrical objects in a complex phase space coming from classical mechanics and correspond to a generalization of Bohr-Sommerfeld quantization conditions in the non self-adjoint case. First, we study non self-adjoint perturbations of self-adjoint pseudo-differential operators in dimension one by using techniques of analytic microlocal analysis. As a corollary, we establish for PT-symmetric perturbations of self-adjoint operators, that the spectrum is real. Then we show Bohr-Sommerfeld quantization conditions for non self-adjoint perturbations of self-adjoint Berezin-Toeplitz operators of the complex plane. In the second part, we look into quantizations of the torus, namely the Berezin-Toeplitz, the classical Weyl and the complex Weyl quantizations of the torus. We establish links between these different quantizations using Bargmann transform. We propose a conjecture, supported by numerical simulations, on Bohr-Sommerfeld quantization conditions for non self-adjoint perturbations of self-adjoint Berezin-Toeplitz operators of the torus.