M.Sc.

M.Sc. (Mathematics) / Any sensible piece of writing has an intended readership. Conversely, any piece of writing that has no intended readership has no sense. These are axioms of authorship and necessary directions to any prospective author. The aim of this dissertation was to serve as an experimental exposition of the analysis of the resolvent operator. Its intended readership is therefore graduate-level students in operator theory and Banach algebras. The analysis included in this dissertation is of a specific kind: it includes and occasionally extends beyond the analysis of a function at certain of its singularities of finite order. The exposition is experimental in the sense that it does not even aim at a comprehensive review of analysis of the resolvent operator, but it is concerned with that part of it which seems to have interesting and useful results and which appears to be the most suggestive of further research. In order to obtain an exhaustive exposition, we still lack a study of the properties of the resolvent operator where it is differentiable (which seemingly entails little more than undergraduate-level complex analysis), and a study of essential singularities of the resolvent operator (which seems too difficult for the expository style). A brief overview of the contents of this dissertation is in order: a chapter introducing some analytic concepts used throughout this dissertation; a chapter on poles of order 1 follows (so-called simple poles), where the Gelfand theorem (2.1.1) is the most important result; a chapter on poles of higher order, where the Hille theorem is the most prominent; and lastly some topics that have arisen out of the study of poles of the resolvent, collected in chapter 4. I should make it abundantly clear to the reader that although this dissertation is my work, it does not for the most part follow that the result are my own. What is my own is the arrangement, but as it is a literature study, the results are mainly those of other authors. My own addition has been mostly notes, usually in italics. The literature study has benefited very much from Zemanek's paper (Zemanek,[54]), and I am deeply indebted to him for it. Incidentally, this has also been a chance to exhibit my style of citation; the number corresponds to the number of the citation in the bibliography. There are numerous instances where I have indicated possible extensions and recumbent studies that could be roused effectively, but which have swelled this volume unnecessarily. For instance, the last subsection is little more than such indications.

We use a resolvent method to study atomic and molecular negative ions in a constant electric field potential which is linear. When a linear potential is applied, it makes the shape of the original potential of the system slanted into one side and thus changes the time evolution of the system. In particular, a bound state can be changed into a state, so called 'quasibound' state, which is not bound anymore and decays into the continuum due to the presence of the linear potential. For an atomic system, we use an attractive delta function fixed at the origin for the interaction potential and solve the single particle Schrodinger equation. For an actual system, we choose the Hydrogen negative ion, and determine the strength of the delta function so that the bound state energy can simulate the electron affinity of the Hydrogen. We find the resolvent of the system and the poles of the resolvent in the analytically continued region. From the patterns of the location of the poles, we can view the one delta function system as a combination of three simple systems. Though they are not exactly the same, this view gives some insight on the system. From the residue at each pole, complex eigenstates are constructed and used for the calculation of the survival probability of an initial state. For the same initial state, we calculate the photodetachment rate when a time-periodic potential is applied. The plot for the photodetachment rate shows peaks at certain incident photon energies. These are compared with an experimental data and give a good agreement although our model is just one dimensional. For a molecular system, two delta function model is suggested by us as an extension of the one delta function model. We find the resolvent of the system and the pole structure from the resolvent. The complex eigenstates are constructed from the residue of the resolvent at each pole. We try to model Oxygen molecular negative ion and determine the strength of the delta function and the distance between the delta functions so that they are consistent with the electron affinity and the internuclear distance of the Oxygen molecule. We also calculate the survival probability and the photodetachment rate of an initial state and find that the plot of the photodetachment rate has similar shape to that of the one delta function model. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Let X be a real Banach space and G1, G2 two nonempty, open and bounded subsets of X such that 0 ∈ G2 and G2 ⊂ G1. The problem (∗) T x + Cx = 0 is considered, where T : X ⊃ D(T) → X is an accretive or monotone operator with 0 ∈ D(T) and T(0) = 0, while C : X ⊃ D(C) → X can be, e.g., one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type (S+) with T positively homogeneous of degree one; (d) quasi-bounded and satisfies a generalized (S+)-condition w.r.t. the operator T, while T is positively homogeneous of degree one. Solutions are sought for the problem (∗) lying in the set D(T + C) ∩ (G1 \ G2). Nontrivial solutions of (∗) exist even when C(0) = 0. The degree theories of Leray and Schauder, Browder, and Skrypnik as well as the degree theory by Kartsatos and Skrypnik for densely defined operators T, C are used. The last three degree theories do not assume any compactness conditions on the operator C. The excision and additivity properties of these degree theories are employed, and the main results are significant extensions or generalizations of previous results by Krasnoselskii, Guo, Ding and Kartsatos involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator T. Moreover, a new degree theory developed by Kartsatos and Skrypnik has been used to prove a similar result for operators of type T + C, where T : X ⊃ D(T) → 2 X∗ is a multi-valued maximal monotone operator, with 0 ∈ D(T) and 0 ∈ T(0), and C : X ⊃ D(C) → X∗ is a densely defined quasi-bounded and finitely continuous operator of type (S˜+). The problem of existence of nonzero solutions for T x + Cx + Gx 3 0 is also considered. Here, T is maximal monotone, C is bounded demicontinuous of type (S+), and G is of class (P). Eigenvalue and invariance of domain results have also been established for the sum L + T + C : G ∩ D(L) → 2 X∗ , where G ⊂ X is open and bounded, L : X ⊃ D(L) → X∗ densely defined linear maximal monotone, T : X → 2X∗ bounded maximal monotone, and C : G → X∗ bounded demicontinuous of type (S+) w. r. t. D(L).

The study of integral operators on spaces of analytic functions has been considered for the past few decades. However, most of the studies in this line are based on spaces of analytic functions of the unit disc. For the analytic spaces of the upper half-plane, the literature is still scanty. Most notable is the recent work of Siskakis and Arvanitidis concerning the classical Ces`aro operator on Hardy spaces of the upper half-plane. In this dissertation, we characterize all continuous one-parameter groups of automorphisms of the upper halfplane according to the nature and location of their fixed points into three distinct classes, namely, the scaling, the translation, and the rotation groups. We then introduce the associated groups of weighted composition operators on both Hardy and weighted Bergman spaces of the half-plane. Interestingly, it turns out that these groups of composition operators form three strongly continuous groups of isometries. A detailed analysis of each of these groups of isometries is carried out. Specifically, we determine the spectral properties of the generators of every group, and using both spectral and semigroup theory of Banach spaces, we obtain concrete representations of the resolvents as integral operators on both Hardy and Bergman spaces of the half-plane. For the scaling group, the resulting resolvent operators are exactly the Ces`aro-like operators. The spectral properties of the obtained integral operators is also determined. Finally, we detail the theory of both Szeg¨o and Bergman projections of the half-plane, and use it to determine the duality properties of these spaces. Consequently, we obtain the adjoints of the resolvent operators on the reflexive Hardy and Bergman spaces of the half-plane.

Distributed optimization over multi-agent networks has become an increasingly popular research topic as it incorporates many applications from various areas such as consensus optimization, distributed control, network resource allocation, large scale machine learning, etc. Parallel distributed solution algorithms are highly desirable as they are more scalable, more robust against agent failure, align more naturally with either underlying agent network topology or big-data parallel computing framework. In this dissertation, we consider a multi-agent optimization formulation where the global objective function is the summation of individual local objective functions with respect to local agents' decision variables of different dimensions, and the constraints include both local private constraints and shared coupling constraints. Employing and extending tools from the monotone operator theory (including resolvent operator, operator splitting, etc.) and fixed point iteration of nonexpansive, averaged operators, a series of distributed solution approaches are proposed, which are all iterative algorithms that rely on parallel agent level local updates and inter-agent coordination. Some of the algorithms require synchronizations across all agents for information exchange during each iteration while others allow asynchrony and delays. The algorithms' convergence to an optimal solution if one exists are established by first characterizing them as fixed point iterations of certain averaged operators under certain carefully designed norms, then showing that the fixed point sets of these averaged operators are exactly the optimal solution set of the original multi-agent optimization problem. The effectiveness and performances of the proposed algorithms are demonstrated and compared through several numerical examples.<br>

Es wird gezeigt, dass ein Satz über die Abbildung spektraler Lücken, welcher exponentielle Dichotomie charakterisiert, für eine allgemeine Klasse (SH) von semilinearen hyperbolischen Systemen von partiellen Differentialgleichungen in einem Banach-Raum X von stetigen Funktionen gilt. Dies beantwortet ein Schlüsselproblem für die Existenz und Glattheit invarianter Mannigfaltigkeiten semilinearer hyperbolischer Systeme. Unter natürlichen Annahmen an die Nichtlinearitäten wird gezeigt, dass schwache Lösungen von (SH) einen glatten Halbfluß im Raum X bilden. Für Linearisierungen werden hochfrequente Abschätzungen für Spektren sowie Resolventen unter Verwendung von reduzierten (block)diagonal Systemen hergestellt. Darauf aufbauend wird der Abbildungssatz für spektrale Lücken im kleinen Raum X bewiesen: Eine offene spektrale Lücke des Generators wird exponentiell auf eine offene spektrale Lücke der Halbruppe abgebildet und umgekehrt. Es folgt, dass ein Phänomen wie im Gegenbeispiel von Renardy nicht auftreten kann. Unter Verwendung der allgemeinen Theorie implizieren die Ergebnisse die Existenz von glatten Zentrumsmannigfaltigkeiten für (SH). Die Ergebnisse werden auf traveling wave Modelle für die Dynamik von Halbleiter Lasern angewandt. Für diese werden Moden Approximationen (Systeme von gewöhnlichen Differentialgleichungen, welche die Dynamik auf gewissen Zentrumsmannigfaltigkeiten approximativ beschreiben) hergeleitet und gerechtfertigt, die generische Bifurkation von modulierten Wellen aus rotierenden Wellen wird gezeigt. Globale Existenz und glatte Abhängigkeit von nichtautonomen traveling wave Modellen werden betrachtet, außerdem werden Moden Approximationen für solche nichtautonomen Modelle rigoros hergeleitet. Insbesondere arbeitet die Theorie für die Stabilitäts- und Bifurkationsanalyse von Turing Modellen mit korellierter Zufallsbewegung. Ferner beinhaltet die Klasse (SH) neutrale und retardierte funktionale Differentialgleichungen. / A spectral gap mapping theorem, which characterizes exponential dichotomy, is proven for a general class of semilinear hyperbolic systems of PDEs in a Banach space X of continuous functions. This resolves a key problem on existence and smoothness of invariant manifolds for semilinear hyperbolic systems. It is shown that weak solutions to (SH) form a smooth semiflow in X under natural conditions on the nonlinearities. For linearizations high frequency estimates of spectra and resolvents in terms of reduced diagonal and blockdiagonal systems are given. Using these estimates a spectral gap mapping theorem in the small Banach space X is proven: An open spectral gap of the generator is mapped exponentially to an open spectral gap of the semigroup and vice versa. Hence, a phenomenon like in Renardy''s counterexample cannot appear for linearizations of (SH). By the general theory the results imply existence of smooth center manifolds for (SH). Moreoever, the results are applied to traveling wave models of semiconductor laser dynamics. For such models mode approximations (ODE systems which approximately describe the dynamics on center manifolds) are derived and justified, and generic bifurcations of modulated waves from rotating waves are shown. Global existence and smooth dependence of nonautonomous traveling wave models with more general solutions, which possess jumps, are considered, and mode approximations are derived for such nonautonomous models. In particular the theory applies to stability and bifurcation analysis for Turing models with correlated random walk. Moreover, the class (SH) includes neutral and retarded functional differential equations.