Browsing by Subject "Banach spaces"
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Item Abstract Riccati equations in a finite Lp1s space and applications to transport theory(Texas Tech University, 1987-08) Juang, JonqNot availableItem Bounded operators without invariant subspaces on certain Banach spaces(2001-12) Jiang, Jiaosheng; Rosenthal, Haskell P.Item Cluster Value Problems in Infinite-Dimensional Spaces(2014-08-05) Ortega Castillo, SofiaIn this dissertation we study cluster value problems for Banach algebras H(B) of analytic functions on the open unit ball B of a Banach space X that contain X* and 1. Solving cluster value problems requires understanding the cluster set of a function f ? H(B). For the Banach spaces X we focus on, such as those with a shrinking reverse monotone Finite Dimensional Decomposition and C(K), we prove cluster value theorems for a Banach algebra H(B) and a point x** ? B ?**. In doing so, we apply standard methods and results in functional analysis; in particular we use the facts that projections from X onto a finite-codimensional subspace equal I_(X) minus a finite rank operator and that C(K)* = l_(1)(K) when K is compact, Hausdorff and dispersed. We also prove that for any separable Banach space Y , a cluster value problem for H(BY ) (H = H? or H = Au) can be reduced to a cluster value problem for H(BX) for some Banach space X that is an l_(1)-sum of a sequence of finite-dimensional spaces. The proof relies on the construction of an isometric quotient map from a suitable X to Y that induces an isometric algebra homomorphism from H(BY ) to H(BX) with norm one left inverse. The left inverse is built using ultrafilter techniques. Other tools include the infinite-dimensional version of the Schwarz lemma and familiar one complex variable results such as Cauchy's inequality and Montel's theorem. We conclude this work by describing the related ? ? problem and defining strong pseudoconvexity as well as uniform strong pseudoconvexity in the context of Banach spaces. Our last result is that 2-uniformly PL-convex Banach spaces have a uniformly strictly pseudoconvex unit ball. In future research we will study the ? ? problem in uniformly strictly pseudoconvex unit balls and in the open unit ball of finite-dimensional Banach spaces such as the ball of l_1^n.Item Commutators on Banach Spaces(2010-10-12) Dosev, DetelinA natural problem that arises in the study of derivations on a Banach algebra is to classify the commutators in the algebra. The problem as stated is too broad and we will only consider the algebra of operators acting on a given Banach space X. In particular, we will focus our attention to the spaces $\lambda I and $\linf$. The main results are that the commutators on $\ell_1$ are the operators not of the form $\lambda I + K$ with $\lambda\neq 0$ and $K$ compact and the operators on $\linf$ which are commutators are those not of the form $\lambda I + S$ with $\lambda\neq 0$ and $S$ strictly singular. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain these results and use this generalization to obtain partial results about the commutators on spaces $\mathcal{X}$ which can be represented as $\displaystyle \mathcal{X}\simeq \left ( \bigoplus_{i=0}^{\infty} \mathcal{X}\right)_{p}$ for some $1\leq p\leq\infty$ or $p=0$. In particular, it is shown that every non - $E$ operator on $L_1$ is a commutator. A characterization of the commutators on $\ell_{p_1}\oplus\ell_{p_2}\oplus\cdots\oplus\ell_{p_n}$ is also given.Item Operator Ideals in Lipschitz and Operator Spaces Categories(2012-10-19) Chavez Dominguez, JavierWe 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.