A pair of commuting bounded operators (S; P ) acting on a Hilbert space, is called a -contraction, if it has the symmetrised bides = f(z1 + z2; z1z2) : jz1j 1; jz2j 1g C2 as a spectral set. For every -contraction (S; P ), the operator equation S S P = DP F DP has a unique solution F 2 B(DP ) with numerical radius, denoted by w(F ), no greater than one, where DP is the positive square root of (I P P ) and DP = RanDP . This unique operator is called the fundamental operator of (S; P ). This thesis constructs an explicit normal boundary dilation for -contractions. A triple of commuting bounded operators (A; B; P ) acting on a Hilbert space with the tetra block E = f(a11; a22; detA) : A = a11 a12 with kAk 1g C 3 a21 a22 as a spectral set, is called a tetra block contraction. Every tetra block contraction possesses two fundamental operators and these are the unique solutions of A B P = DP F1DP ; and B A P = DP F2DP : Moreover, w(F1) and w(F2) are no greater than one. This thesis also constructs an explicit normal boundary dilation for tetra block contractions. In these constructions, the fundamental operators play a pivotal role. Both the dilations in the symmetrised bidisc and in the tetra block are proved to be minimal. But unlike the one variable case, uniqueness of minimal dilations fails in general in several variables, e.g., Ando's dilation is not unique, see [44]. However, we show that the dilations are unique under a certain natural condition. In view of the abundance of operators and their complicated structure, a basic problem in operator theory is to find nice functional models and complete sets of unitary invariants. We develop a functional model theory for a special class of triples of commuting bounded operators associated with the tetra block. We also find a set of complete unitary invariants for this special class. Along the way, we find a Burling-Lax-Halmos type of result for a triple of multiplication operators acting on vector-valued Hardy spaces. In both the model theory and unitary invariance, fundamental operators play a fundamental role. This thesis answers the question when two operators F and G with w(F ) and w(G) no greater than one, are admissible as fundamental operators, in other words, when there exists a -contraction (S; P ) such that F is the fundamental operator of (S; P ) and G is the fundamental operator of (S ; P ). This thesis also answers a similar question in the tetra block setting.

Generalizando para o caso do bidisco hiperbólico as construções em (ANANIN; GROSSI; GUSEVISKII, 2011) e em (GROSSI, 2015), provamos que o fibrado trivial (tangente) sobre superfícies de gênero ≥ 1 (≥ 2) admite geometria modelada no bidisco hiperbólico. (O caso do fibrado trivial sobre o toro é particularmente curioso, pois a curvatura é nula na base e em cada fibra, mas não no fibrado.) Além do seu próprio valor intrínseco, estes exemplos se inserem no contexto da conjectura de Gromov, Lawson e Thurston. Originalmente, a conjectura de Gromov, Lawson e Thurston diz que um fibrado de discos sobre uma superfície conexa fechada orientável de gênero ≥ 2 admite métrica hiperbólica completa de curvatura constante se e só se ΙeΙ ≤ Ι XΙ, onde e é o número de Euler do fibrado e X é a caraterística de Euler da base. Posteriomente, observou-se que esta desigualdade também era válida em todos os fibrados de discos sobre superfícies com estrutura hiperbólica complexa (i.e., uniformizados pela 2-bola holomorfa) conhecidos. Por esta razão, passou-se a acreditar que a conjectura depende apenas de curvatura negativa lato sensu (digamos, à la Alexandrov) e não das especificidades de uma geometria hiperbólica particular. O bidisco hiperbólico é o caso mais simples que nos permite testar tal hipótese, pois está no limite de ser hiperbólico (a curvatura é ≤ 0). Construímos os dois casos extremais: = 0 (fibrado trivial) e ΙeΙ = ΙXΙ (fibrado tangente). Além disso, provamos alguns resultados relacionados à teoria de Teichmüller no contexto de fibrados de discos uniformizados pelo bidisco hiperbólico. / Generalizing the constructions in (ANANIN; GROSSI; GUSEVISKII, 2011) and in (GROSSI, 2015) to the hyperbolic bidisc, we show that the trivial (tangent) bundle over genus ≥ 1 (≥ 2) surfaces admits a geometric structure modelled on the hyperbolic bidisc. (The case of the trivial bundle over the torus is particularly interesting because the curvature vanishes on the base and on every fiber, but is non-null on the bundle.) Aside from their intrinsic value, these examples also play a role in the context of the Gromov, Lawson and Thurston conjecture (GLT conjecture). Originally, the GLT conjecture states that a disc bundle over a connected oriented closed surface of genus ≥ 2 admits a complete hyperbolic metric of constant curvature if and only if ΙeΙ ≤ ΙXΙ, where stands for the Euler number of the bundle and , for the Euler characteristic of the base. Afterwards, it was observed that this inequality also holds for every known example of disc bundles over surfaces equipped with complex hyperbolic structure (i.e., uniformized by the holomoprhic 2-ball). So, one started to believe that the conjecture depends only on negative curvature lato sensu (say, à la Alexandrov) and not on the particularities of an specific hyperbolic geometry. The hyperbolic bidisc is the simplest case allowing us to test such hypothesis since it lies on the frontier of being hyperbolic (curvature is ≥ 0). We construct the two extremal cases: e = 0 (trivial bundle) and ΙeΙ = ΙXΙ (tangent bundle). We also prove a few results related to Teichmüllers theory in the context of disc bundles uniformized by the hyperbolic bidisc.

Generalizando para o caso do bidisco hiperbólico as construções em (ANANIN; GROSSI; GUSEVISKII, 2011) e em (GROSSI, 2015), provamos que o fibrado trivial (tangente) sobre superfícies de gênero ≥ 1 (≥ 2) admite geometria modelada no bidisco hiperbólico. (O caso do fibrado trivial sobre o toro é particularmente curioso, pois a curvatura é nula na base e em cada fibra, mas não no fibrado.) Além do seu próprio valor intrínseco, estes exemplos se inserem no contexto da conjectura de Gromov, Lawson e Thurston. Originalmente, a conjectura de Gromov, Lawson e Thurston diz que um fibrado de discos sobre uma superfície conexa fechada orientável de gênero ≥ 2 admite métrica hiperbólica completa de curvatura constante se e só se ΙeΙ ≤ Ι XΙ, onde e é o número de Euler do fibrado e X é a caraterística de Euler da base. Posteriomente, observou-se que esta desigualdade também era válida em todos os fibrados de discos sobre superfícies com estrutura hiperbólica complexa (i.e., uniformizados pela 2-bola holomorfa) conhecidos. Por esta razão, passou-se a acreditar que a conjectura depende apenas de curvatura negativa lato sensu (digamos, à la Alexandrov) e não das especificidades de uma geometria hiperbólica particular. O bidisco hiperbólico é o caso mais simples que nos permite testar tal hipótese, pois está no limite de ser hiperbólico (a curvatura é ≤ 0). Construímos os dois casos extremais: = 0 (fibrado trivial) e ΙeΙ = ΙXΙ (fibrado tangente). Além disso, provamos alguns resultados relacionados à teoria de Teichmüller no contexto de fibrados de discos uniformizados pelo bidisco hiperbólico. / Generalizing the constructions in (ANANIN; GROSSI; GUSEVISKII, 2011) and in (GROSSI, 2015) to the hyperbolic bidisc, we show that the trivial (tangent) bundle over genus ≥ 1 (≥ 2) surfaces admits a geometric structure modelled on the hyperbolic bidisc. (The case of the trivial bundle over the torus is particularly interesting because the curvature vanishes on the base and on every fiber, but is non-null on the bundle.) Aside from their intrinsic value, these examples also play a role in the context of the Gromov, Lawson and Thurston conjecture (GLT conjecture). Originally, the GLT conjecture states that a disc bundle over a connected oriented closed surface of genus ≥ 2 admits a complete hyperbolic metric of constant curvature if and only if ΙeΙ ≤ ΙXΙ, where stands for the Euler number of the bundle and , for the Euler characteristic of the base. Afterwards, it was observed that this inequality also holds for every known example of disc bundles over surfaces equipped with complex hyperbolic structure (i.e., uniformized by the holomoprhic 2-ball). So, one started to believe that the conjecture depends only on negative curvature lato sensu (say, à la Alexandrov) and not on the particularities of an specific hyperbolic geometry. The hyperbolic bidisc is the simplest case allowing us to test such hypothesis since it lies on the frontier of being hyperbolic (curvature is ≥ 0). We construct the two extremal cases: e = 0 (trivial bundle) and ΙeΙ = ΙXΙ (tangent bundle). We also prove a few results related to Teichmüllers theory in the context of disc bundles uniformized by the hyperbolic bidisc.

La nitruration gazeuse est utilisée par les fabricants d’engrenages pour traiter la surface des dentures. En effet, l’augmentation de la dureté et l’introduction de contraintes résiduelles de compression contribuent à retarder voire inhiber l’initiation et la propagation des fissures superficielles de fatigue de contact. Cependant la majorité des aciers alliés présentent des réseaux de précipités de carbures dans la couche nitrurée. Ceux-ci ont la particularité de se retrouver aux joints de grains relativement parallèles à la surface, ce qui leurs ont attribué les surnoms de «liserés de carbures». Associés à une phase dure et fragile, ces carbures constituent des zones d’hétérogénéités. Par ailleurs, la norme pour la qualité des matériaux d’engrenages autorise une large gamme de tailles de grains, conduisant à diverses microstructures pour les couches nitrurées des aciers alliés. En effet, la morphologie des réseaux de carbures après nitruration dépend entre autres de la taille de grain de l’acier. De par le manque de travaux sur l’impact réel de ces précipités sur la fatigue de contact, la présente étude propose des analyses d’essais. Les essais consistent à reproduire de l’écaillage initié en surface sur des éprouvettes dont les couches nitrurées, issues d’un seul traitement thermochimique, présentent les mêmes propriétés mécaniques (dureté et contraintes résiduelles) mais différentes propriétés microstructurales (tailles de grains et morphologies de carbures). Les analyses ont permis de proposer des mécanismes d’initiation de micro-fissures induites en proche surface sous les conditions d’essai appliquées. Les sites d’initiation dépendent de la taille de grain de l’acier et peuvent en particulier s’établir au niveau des carbures intergranulaires. L’étude des réseaux de fissures couplée à l’analyse des contraintes résiduelles a permis d’établir des scénarii de propagation des fissures dans les couches nitrurées. Sous les conditions de contact appliquées, l’apparition de déformation plastique entraîne, par accommodation des variations de volume, la relaxation des contraintes résiduelles compressives initialement présentes dans le matériau, aux profondeurs les moins sollicitées. Ces contraintes étant moins compressives, elles libèrent les micro-fissures déjà présentes en proche surface. Dès lors, la propagation au travers des carbures qui s’apparentent à des sites privilégiés pour la croissance des fissures en cours de fatigue est possible. En particulier, ces précipités entraînent les fissures vers le cœur lorsqu’ils sont en forte densité dans les couches nitrurées, tandis que l’endommagement reste superficiel lorsque l’éloignement entre les précipités est trop important. Enfin, la continuité et la longueur des carbures augmentent localement la vitesse de propagation des fissures. Ceci se traduit par l’apparition plus rapide de l’écaillage initié en surface pour les couches nitrurées de microstructures grossières. / Gas nitriding is a thermochemical surface treatment widely used by gear manufacturers to improve the rolling contact fatigue endurance of their components. Indeed, increasing the hardness and introducing compressive residual stresses to the geartooth surfaces inhibit or delay crack initiation and propagation. However, most of the alloyed steel nitrided layers show the presence of intergranular carbide networks. The precipitation of these carbides specifically occurs at parallel grain boundaries during the treatment and, being a carbon-rich phase, they are associated to tough and fragile heterogeneities. Besides, standards for gear material quality recommend a broad range of steel grain sizes which lead to various possible microstructures on gear components after nitriding. Indeed, the intergranular carbide network morphology depends on the steel grain size. Since there is no evidence regarding the real effect of the carbides on rolling contact fatigue, this work presents experimental investigations carried out on a twin-disc machine. The tests consist in reproducing surface-initiated pitting on specimens whose nitrided layers, obtained by a unique thermochemical surface treatment, display similar mechanical properties (hardness and compressive residual stresses) but different microstructures (grain sizes and carbide network morphologies). The first analysis resulted in suggesting the micro-crack nucleation mechanisms. In this study, the test conditions were chosen to induce these micro-cracks near the disc surfaces. For a given contact stress field, nucleation sites depend on the steel grain size and can take place at intergranular carbides. Investigations of crack networks led on cross sections and 3D observations by means of high energy X-ray computed tomography, coupled with compressive residual stress evolution analysis, help the authors proposing the rolling contact fatigue crack propagation scenarios in nitrided layers. When contact shear stresses locally exceed the material micro-yield shear stress limit, micro-deformations arise and add compressive residual stresses to the treated layers. To accommodate these volume variations the compressive residual stresses, initially induced by nitriding, release at depths where contact stresses are lower. The pre-existing residual stresses being less compressive, micro-cracks near the surface can propagate through the carbides that act as preferential crack growth sites. When the nitrided layers display high carbide density, the intergranular precipitates drag the cracks toward the core, whereas rolling contact fatigue failures are limited to the near surface when the distance between the precipitates is too important. Finally, the length and the continuity of the carbides, linked to the steel grain size, locally increase the crack propagation speed. This gives rise to a lower endurance to surface-initiated pitting in coarse microstructures.