Optimal L^2 Bounds for Certain Hamiltonian Linearizations and New Generalizations of Asymptotic Structures in Certain Banach Spaces

Abstract

This thesis is the culmination of two distinct branches of research during my time as a graduate student. First, with my advisor Milena Stanislavova, I studied the theory of one-parameter semigroups of bounded linear operators acting on a Banach space. We used aspects of this theory to derive optimal (depending on the eigenvalue structure) $L^{2}$ bounds on the semigroup solutions to certain Hamiltonian linearized partial differential equations when it is assumed that the spectrum of the Hamiltonian linearized operator is purely imaginary. Without this assumption, our methods allowed us still to infer a priori bounds on spectrum of the linearized operator. The first chapter of this thesis is an introduction to semigroup theory and the second chapter details how we applied these methods (specifically, the Gomilko Lemma which allows us to distinguish uniform and exponential decay for the semigroup solution) to certain Hamiltonian linearized nonlinear Schr\'{o}dinger and Korteweg-De Vries equations. Next, I pursued independently a research project in the geometry of Banach spaces. A Banach space $X$ is said to have the Property of Lebesgue or to be a ``PL-space" if every Riemann-integrable function $f:[0,1]\to X$ is Lebesgue almost everywhere continuous. The problem of characterizing PL-spaces in terms of their asymptotic geometry is still open. I believe that the solution to this problem will come with the advent of stronger local results which more easily allow for the inference of some amount of global asymptotic structure of $X$. Upgrading local asymptotic results to global ones is in general a difficult and ongoing problem in the geometry of Banach spaces. Whether or not $X$ is a PL-space is intimately linked to its asymptotic proximity (both global and local) to $\ell_{1}$. In 2008, K.M. Naralenkov proved that every Banach space that is asymptotic-$\ell_{1}$ with respect to a basis is a PL-space and he also provided details to an unpublished result of A. Pelczy\'{n}ski and G.C. da Rocha Filho that every spreading model of a PL-space is equivalent to $\ell_{1}$. I generalized both of these results in my recent paper \cite{Gaebler} so that every Banach space that is asymptotic-$\ell_{1}$ in a coordinate-free sense (i.e. it need not have a basis so this would include non-separable spaces) is a PL-space, and every so-called SP-asymptotic model of a PL-space is equivalent to $\ell_{1}$. SP-asymptotic models directly generalize spreading models, and the few local-to-global results that do exist are often framed in terms of asymptotic models. The last chapter of this thesis provides some background about PL-spaces, details these new contributions to their theory, and closes with some specific ideas for future research on this topic.