Cluster Value Problems in Infinite-Dimensional Spaces

Abstract

In 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.