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.