Orientadores: Pedro José Catuogno, Johannes Michael Antonius Maria van Neerven / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T15:55:30Z (GMT). No. of bitstreams: 1 AbreuJunior_JamilGomesde_D.pdf: 1681574 bytes, checksum: 280ab5f7ecf646a3ab11f04ca34664e3 (MD5) Previous issue date: 2013 / Resumo: Nesta tese tratamos o problema de caracterizar a existência de medida invariante para equações de evolução estocásticas lineares com ruído aditivo em termos do resolvente associado ao gerador da equação. Este problema foi proposto recentemente na literatura como uma versão estocástica da célebre conjectura de Weiss em teoria de controle para sistemas lineares, que consiste em relacionar admissibilidade de operadores de controle a certas estimativas envolvendo o resolvente do gerador infinitesimal. No contexto estocástico, e no caso em que o gerador da equação é analítico e admite um cálculo funcional do tipo Dunford-Schwartz num espaço de Banach com a propriedade de Pisier, nosso resultado principal consiste de condições analítico-funcionais necessárias e suficientes para existência de medida invariante para o problema de Cauchy estocástico. Em particular, mostramos que existência de medida invariante _e equivalente _a convergência em probabilidade de certa série Gaussiana cujos termos são os resolventes avaliados nos pontos diádicos positivos da reta real, que consideramos como sendo a condição de Weiss estocástica. Há fortes razões para esperar que, _a semelhança do que ocorreu com a conjectura de Weiss clássica, este problema atraia considerável atenção da comunidade acadêmica num futuro próximo / Abstract: In this thesis we consider the problem of characterizing the existence of invariant measure for linear stochastic evolution equations with additive noise in terms of the resolvent operator associated to the generator of the equation. This problem was recently proposed in the literature as a stochastic version of the celebrated Weiss conjecture in linear systems theory, which relates admissibility of control operators to certain estimates involving the resolvent of the infinitesimal generator. In the stochastic setting and when the generator is analytic and admits a bounded functional calculus in a Banach space with Pisier property, our main result consists of necessary and sufficient functional analytic conditions for the existence of an invariant measure for the stochastic Cauchy problem. In particular, we show that existence of invariant measure is equivalent to convergence in probability of a certain Gaussian series whose terms are the resolvents evaluated at the positive dyadic points of the real line, which we consider as being the stochastic Weiss condition. There are strong reasons to expect that, similarly to what happened to the classical Weiss conjecture, this work will attract considerable attention of the academic community in the near future / Doutorado / Matematica / Doutor em Matemática

Orientador: Lucas Catão de Freitas Ferreira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T19:21:09Z (GMT). No. of bitstreams: 1 ValenciaGuevara_JulioCesar_M.pdf: 664823 bytes, checksum: 0c43bba776e592ed44fad0d1bc2f6998 (MD5) Previous issue date: 2012 / Resumo: Estudamos existência, unicidade, dependência contínua nos dados e comportamento assint ótico de soluções globais das equações de Navier-Stokes (com n >= 3), sob condições de pequenez no dado inicial e na força externa, em um espaço de distribuições (PMa) cuja construção é baseada na transformada de Fourier. Este espaço contém funções fortemente singulares e, em particular, funções homogêneas de um certo grau cuja correspondente solução (com tais dados) é auto-similar. Além disso, mostramos a existência de uma classe de soluções que são assintoticamente auto-similar. Estudamos também a existência de soluções estacionárias pequenas e analisamos a estabilidade assintótica delas. Finalmente, são dadas condições sob as quais a solução é uma função regular para t > 0 (mesmo com dado inicial singular) e satisfaz as equações de Navier-Stokes no sentido clássico para t > 0. Esta dissertação é baseada no artigo de M. Cannone and G. Karch, Journal of Diff. Equations 197 (2) (2004) / Abstract: We study existence, uniqueness, continuous dependence upon the data and asymptotic behavior of solutions for the Navier-Stokes equations (with n _ 3), under smallness conditions on the initial data and external force, in a space of distributions (PMa), whose construction is based on Fourier transform. This space contains strongly singular functions and, in particular, homogeneous functions with a certain degree whose corresponding solution (with such data) is self-similar. Moreover, the existence of a class of asymptotically self-similar solutions is proved. We also study the existence of small stationary solutions and their asymptotic stability. Finally, conditions are given for the obtained solution to be regular for t > 0 (even with singular initial data) and to satisfy the Navier-Stokes equations in the classical sense for t > 0. This master dissertation is based on the paper by M. Cannone and G. Karch, Journal of Diff. Equations 197 (2) (2004) / Mestrado / Matematica / Mestre em Matemática

Orientador: Jorge Tulio Mujica Ascui / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T12:08:37Z (GMT). No. of bitstreams: 1 MelendezCaraballo_Blas_M.pdf: 1140173 bytes, checksum: 61bc3f801fdfc8946dd6852692a39bfd (MD5) Previous issue date: 2015 / Resumo: Em 1960, Pelczynski [1] provou que, se X é um dos espaços c0 ou lp, com p número real maior ou igual do que um. Então todo subespaço complementado de dimensão infinita de X é isomorfo a X. Outro resultado clássico de Pelczynski [1] afirma que se p é um número real maior do que um, então o espaço Lp[0,1] contém um subespaço complementado isomorfo a l2. Nosso objetivo é estudar os resultados deste tipo, e introduzir alguns problemas abertos. BIBLIOGRAFIA [1] A. Pelczynski, Projections in certain Banach spaces, Studia Methematica, 19 (1960), pág. 209-228 / Abstract: In 1960, Pelczynski [1] showed that if X is one of the spaces c0 or lp, p real number greater than or equal to one. Then each infinite dimensional subspace complemented in X is isomorphic to X. Another classical result of Pelczynski [1] states that if p is a real number greater that one, then the space Lp[0,1] contains a complemented subspace isomorphic to l2. Our aim is to study results of this kind, and to introduce some open problems. BIBLIOGRAFIA [1] A. Pelczynski, Projections in certain Banach spaces, Studia Methematica, 19 (1960), pág. 209-228 / Mestrado / Matematica / Mestre em Matemática

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Thesis (MSc)--Stellenbosch University, 2013. / Please refer to full text to view abstract.

The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space $XK$ possesses a number of additional interesting properties, most notably, it has $ell_1$ dual. Since $ell_1$ possesses the Schur property, weakly compact and norm compact operators on $XK$ coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that $XK$ has very few operators made no use of the Schur property of $ell_1$. We therefore arrive at the following question (originally posed in cite{AH}): must a HI, $mathcal{L}_{infty}$, $ell_1$ predual with few operators (every operator is a strictly singular perturbation of $lambda I$) necessarily have very few operators? We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $delta$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $ell_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant $ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.

Let X be a Banach space and denote by SS₁(X) the set of all S₁-strictly singular operators from X to X. We prove that there is a Banach space X such that SS₁(X) is not a closed ideal. More specifically, we construct space X and operators T₁ and T₂ in SS₁(X) such that T₁+T₂ is not in SS₁(X). We show one example where the space X is reflexive and other where it is c₀-saturated. We also develop some results about S_alpha-strictly singular operators for alpha less than omega_1. / text

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Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c'est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires

Student Number : 0413210T - MSc dissertation - School of Mathematics - Faculty of Science / The aim of this work is to do a literature study on spaces of martingales on Riesz spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure. Convergence of martingales in the classical theory of stochastic processes has many applications in mathematics and related areas. Operator theoretic approaches to the classical theory of stochastic processes and martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15, 26, 27]. The classical theory of stochastic processes for scalar-valued measurable functions on a probability space ( ,#6;, μ) utilizes the measure space ( ,#6;, μ), the norm structure of the associated Lp(μ)-spaces as well as the order structure of these spaces. Motivated by the existing operator theoretic approaches to classical stochastic processes, a theory of discrete-time stochastic processes has been developed in [16, 19, 20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach is measure-free and utilizes only the order structure of the given Riesz space. Martingale convergence in the Riesz space setting is considered in [18]. It was shown there that the spaces of order bounded martingales and order convergent martingales, on a Dedekind complete Riesz space with a weak order unit, coincide. A measure-free approach to martingale theory on Banach lattices with quasi-interior points has been given in [2]. Here, the groundwork was done to generalize the notion of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space and a Banach lattice (see [1]). In [30], a measure-free approaches to martingale theory on Banach lattices is given. The main results in [30] show that the space of regular norm bounded martingales and the space of norm bounded martingales on a Banach lattice E are Banach lattices in a natural way provided that, for the former, E is an order continuous Banach lattice, and for the latter, E is a KB-space. The definition of a ”martingale” defined on a particular space depends on the type of space under consideration and on the ”filtration,” which is a sequence of operators defined on the space. Throughout this dissertation, we shall consider Riesz spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be determined by the type of space under consideration and will be adapted to suit the case at hand. In Chapter 2, we consider convergent martingale theory on Riesz spaces. This chapter is based on the theory of martingales and their properties on Dedekind complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18]. The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space of martingales with respect to a given filtration on a Riesz space is introduced and an ordering defined on this space. The spaces of regular, order bounded, order convergent and generated martingales are introduced and properties of these spaces are considered. In particular, we show that the space of regular martingales defined on a Dedekind complete Riesz space is again a Riesz space. This result, in this context, we believe is new. The contents of Chapter 3 is convergent martingale theory on Banach lattices. We consider the spaces of norm bounded, norm convergent and regular norm bounded martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E which satisfy the condition 1[n=1 R(Tn) = E, where R(Tn) denotes the range of the filtration, are considered. We do not make this assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition yields equality (in fact, a Riesz and isometric isomorphism) between the space of norm convergent martingales and 1Sn=1R(Tn). The aforementioned main results in [30] are also considered in this chapter. All the results pertaining to martingales on Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new. Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]), on its extension to the M-tensor product of a Banach space and a Banach lattice as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on tensor products of Banach lattices and Banach spaces as can be found in [2]. We show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a Banach space X, then 1[n=1 R(Tn Sn) = 1[n=1 R(Tn) e M 1[n=1 R(Sn). This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales and apply our results to characterize dual Banach spaces with the Radon- Nikod´ym property. We use standard notation and terminology as can be found in standard works on Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However, for the convenience of the reader, notation and terminology used are included in the Appendix at the end of this work. We hope that this will enhance the pace of readability for those familiar with these standard notions.