We study analogues, in the Lipschitz and Operator Spaces categories, of several classical ideals of operators between Banach spaces. We introduce the concept of a Banach-space-valued molecule, which is used to develop a duality theory for several nonlinear ideals of operators including the ideal of Lipschitz p-summing operators and the ideal of factorization through a subset of a Hilbert space. We prove metric characterizations of p-convex operators, and also of those with Rademacher type and cotype. Lipschitz versions of p-convex and p-concave operators are also considered. We introduce the ideal of Lipschitz (q,p)-mixing operators, of which we prove several characterizations and give applications. Finally the ideal of completely (q,p)-mixing maps between operator spaces is studied, and several characterizations are given. They are used to prove an operator space version of Pietsch's composition theorem for p-summing operators.

In this dissertation, we study about the extension of results of psumming operators to Lipschitz p-summing maps and their respective relatives for 1 ≤ p < ∞ . Lipschitz p-summing and Lipschitz p-integral maps are the nonlinear version of (absolutely) p-summing and p-integral operators respectively. The p-summing operators were first introduced in the paper [13] by Pietsch in 1967 for 1 < p < ∞ and for p = 1 go back to Grothendieck which he introduced in his paper [9] in 1956. They were subsequently taken on with applications in 1968 by Lindenstrauss and Pelczynski as contained in [12] and these early developments of the subject are meticulously presented in [6] by Diestel et al. While the absolutely summing operators (and their relatives, the integral operators) constitute important ideals of operators used in the study of the geometric structure theory of Banach spaces and their applications to other areas such as Harmonic analysis, their confinement to linear theory has been found to be too limiting. The paper [8] by Farmer and Johnson is an attempt by the authors to extend known useful results to the non-linear theory and their first interface in this case has appealed to the uniform theory, and in particular to the theory of Lipschitz functions between Banach spaces. We find analogues for p-summing and p-integral operators for 1 ≤ p < ∞. This then divides the dissertation into two parts. In the first part, we consider results on Lipschitz p-summing maps. An application of Bourgain’s result as found in [2] proves that a map from a metric space X into ℓ2X 1 with |X| = n is Lipschitz 1-summing. We also apply the non-linear form of Grothendieck’s Theorem to prove that a map from the space of continuous real-valued functions on [0, 1] into a Hilbert space is Lipschitz p-summing for some 1 ≤ p < ∞. We also prove an analogue of the 2-Summing Extension Theorem in the non-linear setting as found in [6] by showing that every Lipschiz 2-summing map admits a Lipschiz 2-summing extension. When X is a separable Banach space which has a subspace isomorphic to ℓ1, we show that there is a Lipschitz p-summing map from X into R2 for 2 ≤ p < ∞ whose range contains a closed set with empty interior. Finally, we prove that if a finite metric space X of cardinality 2k is of supremal metric type 1, then every Lipschitz map from X into a Hilbert space is Lipschitz p-summing for some 1 ≤ p < ∞. In the second part, we look at results on Lipschitz p-integral maps. The main result is that the natural inclusion map from ℓ1 into ℓ2 is Lipschitz 1-summing but not Lipschitz 1-integral. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted

Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-05-02T19:03:09Z No. of bitstreams: 1 Arquivototal.pdf: 737510 bytes, checksum: a205d9714f9ec661929aea54c8a55145 (MD5) / Made available in DSpace on 2018-05-02T19:03:09Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 737510 bytes, checksum: a205d9714f9ec661929aea54c8a55145 (MD5) Previous issue date: 2017-08-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The main goal of this work is to study a new sequence space introduced in 2014 by Karn and Sinha, namely the space of mid p-summable sequences. More speci cally, we will study a recent work by G. Botelho and J.R. Campos, which deepens the seminal study of this space and presents new classes of operators involving the new space and the classical sequence spaces of absolutely and weakly p-summable sequences, called absolutely mid p-summing and weakly mid p-summing operators. From this, we study a new factorization theorem, involving these new classes of operators, for the absolutely p-summing operators. / O principal objetivo desta dissertação é estudar um novo espaço de sequências introduzido por Karn e Sinha em 2014, a saber, o espaçoo das sequências mid p-somáveis. Mais especi camente, estudaremos um recente trabalho de G. Botelho e J. R. Campos que aprofunda o estudo seminal do espa co e apresenta novas classes de operadores envolvendo este novo espa co e os espa cos cl assicos de sequ^encias absolutamente e fracamente p-somáveis, denominados operadores absolutamente mid p-somantes e operadores fracamente mid p-somantes. A partir disto, estudamos um novo teorema de fatoração, envolvendo estas novas classes de operadores, para os operadores absolutamente p-somantes. mid p-somáveis; Operadores absolutamente e fracamente mid p-somantes.

Made available in DSpace on 2015-05-15T11:46:01Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1921576 bytes, checksum: 089bb16dfc8ec52a2ca6e0e4fea4e360 (MD5) Previous issue date: 2011-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we present some properties of the class of multiple summing multilinear operators. We summarize the theory with the aim of showing in details recent results such as coincidence results, inclusion results and results involving cotype. / No presente trabalho, estudamos algumas propriedades dos operadores multilineares múltiplo somantes. Fazemos um resumo da teoria com o objetivo de apresentar com detalhes resultados recentes de coincidência, inclusão e resultados envolvendo cotipo.

Made available in DSpace on 2015-05-15T11:46:25Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 489943 bytes, checksum: 8456425550cf8ae40851ce99110c0cb5 (MD5) Previous issue date: 2010-12-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present some results involving lineability and the linear theory of absolutely summing operators. We note that the technique presented is closely related to the theory of hereditarily indecomposable Banach spaces and that the presence of an unconditional basis in one of the spaces involved is crucial to guarantee some results. / Neste trabalho apresentamos alguns resultados envolvendo lineabilidade e a teoria linear dos operadores absolutamente somantes. Observamos que a técnica utilizada está intimamente relacionada com a teoria dos espaços de Banach hereditariamente indecomponíveis e que a presença de uma base incondicional em um dos espaços envolvidos é crucial para garantirmos alguns resultados.

Cette thèse est consacrée à l’extension au cadre Banachique de la notion d’opérateur de Hilbert-Schmidt. Dans un premier temps, on étudie d’une part les opérateurs p-sommants dans un espace de Banach X vers un autre espace de Banach Y et d’autre part, les opérateurs gamma-radonifiants dans un espace de Hilbert vers un autre espace de Banach.Dans un second temps, on s'intéresse aux opérateurs gamma-sommants dans des espaces de Banach, qui coïncident avec les opérateurs de Rademacher-bornés, ce qui nous amène aux opérateurs presque sommants. Enfin, on en déduit plusieurs généralisations naturelles de la notion d’opérateur de Hilbert-Schmidt aux espaces de Banach.-Les classes des opérateurs p-sommants de X dans Y .-La classe des opérateurs presque sommants de X dans Y qui coïncide avec la classe des opérateurs gamma-radonifiants de X dans Y.-La classe des opérateurs faible* 1-nucléaires de X dans Y. / This thesis is devoted to extending the notion of Banach Hilbert-Schmidt operator to the framework of Banach spaces. In a first step, we study p-summing operators from a Banach space X into a Banach space Y and gamma-radoniyfing operators from a Hilbert space into a Banach space. In a second step, we discuss gamma-summing operators between Banach spaces, which coincide with Rademacher-bounded operators, which leads to the notion of almost summing operators. Finally, we present serval natural generalizations of the notion of Hilbert-Schmidt operator to Banach spaces.- Classes of p-summing operators from X into Y. - The class of almost summing operators from X into Y, which coincides with the class of gamma-radoniyfing operators from X into Y.- The class of weak*1-nuclear operators from X into Y.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-22T16:03:10Z No. of bitstreams: 1 arquivototal.pdf: 1039820 bytes, checksum: 2e99f469c22f0b9c57e0059499fc3b27 (MD5) / Made available in DSpace on 2017-08-22T16:03:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1039820 bytes, checksum: 2e99f469c22f0b9c57e0059499fc3b27 (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The goal of our work is to study the class of the Cohen strongly summing operators. Initially, we present basic results from Functional Analysis that are necessary for the development of the text and then we deal with sequence spaces which will be used to de ne and study the classes of operators involved in this work, as necessarily the class of the absolutely summing operators. We also study the sequence space of the Cohen- Khalil strongly (q; p)-summable sequences and the sequence space of the Cohen strongly p-summable sequences, as a particular instance of the former. From this, we de ne the class of the Cohen strongly p-summing operators and the class of the Cohen-Khalil strongly (s; r; p)-summing operators which, under certain conditions, are equivalent. We conclude with a study, from the viewpoint of the operator ideal theory, using the abstract environment created by G. Botelho and J. R. Campos, in order to show that p and Dp are Banach ideals and the relations dual p = Dp and Ddual p = p are valid, where p and p are conjugate indexes. / objetivo de nosso trabalho e estudar a classe dos operadores Cohen fortemente p- somantes. Inicialmente, apresentamos resultados b asicos de An alise Funcional necess arios ao desenvolvimento do texto e, em seguida, tratamos dos espa cos de sequ^encias que ser~ao usados na de ni c~ao e estudo das classes de operadores envolvidas no trabalho, como necessariamente a classe dos operadores absolutamente somantes. Apresentamos tamb em o espa co das sequ^encias Cohen-Khalil fortemente (q; p)-som aveis e o espa co das sequ^encias Cohen fortemente p-som aveis, como caso particular do primeiro. A partir disto, de - nimos a classe dos operadores Cohen fortemente p-somantes e a classe dos operadores Cohen-Khalil fortemente (s; r; p)-somantes que, sob certas condi c~oes, s~ao equivalentes. Conclu mos com um estudo, sob o ponto de vista da teoria dos ideais de operadores, usando o ambiente abstrato criado por G. Botelho e J. R. Campos, para mostrar que p e Dp s~ao ideais de Banach e valem as rela c~oes dual p = Dp e Ddual p = p, onde p e p s~ao ndices conjugados.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-16T13:04:43Z No. of bitstreams: 1 arquivototal.pdf: 1098275 bytes, checksum: 59e9bc1ca11ff8f14d5629548ccaaaf8 (MD5) / Made available in DSpace on 2017-08-16T13:04:43Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1098275 bytes, checksum: 59e9bc1ca11ff8f14d5629548ccaaaf8 (MD5) Previous issue date: 2016-03-04 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we study a general version of the Pietsch Domination Theorem, due to Pellegrino, Santos and Seoane-Sep´ulveda, that improves the unified version present in [3] and recovers known Pietsch Domination-type theorems where the unified approach seems not to work. / Neste trabalho estudamos uma vers˜ao geral do Teorema da Domina¸c˜ao de Pietsch, devido a Pellegrino, Santos e Seoane-Sep´ulveda, que melhora a vers˜ao unificada presente em [3] e recupera conhecidos teoremas de domina¸c˜ao do tipo Pietsch onde a abordagem unificada parece n˜ao funcionar.

Made available in DSpace on 2015-05-15T11:46:01Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1973222 bytes, checksum: 2211069e3e843d6b7636b5e87d4ea973 (MD5) Previous issue date: 2011-07-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Bohnenblust-Hille Theorem, proved in 1931 in the prestigious journal Annals of Mathematics, asserts that if U : lN 1 ----- lN 1 --! K is an n-linear form and N is a positive integer N, then 0@ N X i1;:::;in=1 jU(ei1 ; :::; ein)j 2n n+11A n+1 2n - Cn kUk , with Cn = n n+1 2n 2 n--1 2 . After a long time overlooked, this result has been explored in the recent years. In this work we detail a beautiful proof of the Bohnenblust-Hille Theorem, due to A. Defant, U. Schwarting and D. Popa. We also investigate the estimates of the constants involved and some asymptotic information, following a recent work of D. Pellegrino and J. Seoane-Sepúlveda. / O Teorema de Bohnenblust-Hille, demonstrado em 1931 no prestigioso jornal Annals of Mathematics, garante que para toda forma n-linear U : lN 1 - - - - lN 1--! K e para qualquer inteiro positivo N, tem-se - - - - - - - - - - - - - - - - 2 . Após um longo tempo esquecido, esse resultado tem sido bastante explorado nos últimos anos. Neste trabalho fazemos, com detalhes, uma bela demonstração do Teorema de Bohnenblust-Hille, devida a A. Defant, U. Schwarting e D. Popa. Também destacamos o cálculo de estimativas das constantes envolvidas e algumas informações assintóticas, de acordo com um recente trabalho de D. Pellegrino e J. Seoane-Sepúlveda.

Made available in DSpace on 2015-05-15T11:46:05Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 865721 bytes, checksum: 3cb3fb14f515822a2db03f7945b5427a (MD5) Previous issue date: 2013-04-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work presents a study of Cohen strongly summing operators under the viewpoint of the theory of multilinear operators ideals and polynomial ideals. Furthermore, we introduce two new classes that generalize the concept of multilinear operators and polynomials of this nature, namely multiple Cohen strongly summing operators and Cohen strongly summing operators at a given point. We show that the new classes defined, as well as the previous classes, form normed ideals of operators/polynomials and that the class of multiple Cohen strongly summing operators forms a Banach ideal. We also show that the construction of the class of multiple Cohen strongly summing operators provides a holomorphy type and a coherent and compatible sequence of ideals. / Neste trabalho apresentamos um estudo dos operadores Cohen fortemente somantes sob o ponto de vista da teoria de ideais de operadores e polinômios. Além disso, introduzimos duas novas classes de operadores que generalizam o conceito de operadores multilineares e polinômios desta natureza, a saber, os operadores múltiplo Cohen fortemente somantes e os operadores Cohen fortemente somantes num dado ponto. Mostramos que as novas classes definidas, como as anteriores, formam ideais normados de operadores/polinômios e que os operadores múltiplo Cohen fortemente somantes formam um ideal de Banach. Também mostramos que a construção da classe dos operadores múltiplo Cohen fortemente somantes fornece um tipo de holomorfia e uma sequência coerente e compatível de ideais.