Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 112).

In the classical theory of operators on a Banach space a beautiful interplay exists between Riesz and Fredholm theory, and the theory of traces and de¬terminants for operator ideals. In this thesis we obtain a complete Riesz de¬composition theorem for Riesz elements in a semi prime Banach algebra and on the other hand extend the existing theory of traces and determinants to a more general setting of Banach algebras. In order to obtain some of these results we use the notion of finite multiplicity of spectral points to give a characterization of the essential spec¬trum for elements in a Banach algebra. As an immediate corollary we obtain the well-known characterization of Riesz elements namely that their non-zero spectral points are isolated and of finite multiplicities. In the final chapter of the thesis we use Plemelj's type formulas to define a determinant on the ideal of finite rank elements and show that it extends continuously to the ideal of nuclear elements. / Thesis (PhD (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

Associated to a split, semisimple linear algebraic group G is a group of invariant quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic forms in characters of a maximal torus which are fixed with respect to the action of the Weyl group of G. We compute Q(G) for various examples of products of the special linear, special orthogonal, and symplectic groups as well as for quotients of those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Z^n for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces.

Let A be a von Neumann algebra of linear operators on the Hilbert space H . A linear operator T (resp. a linear bounded. functional ϕ ) on A is said to be normal if for any increasing net [formula omitted] of positive elements in A with least upper bound B , T(B) is the least upper bound of [formula omitted]. Two linear positive functionals ψ1 and ψ2 on A are said to be equivalent if ψ1 (B) = 0 <=> ψ2 (B) = 0 for any positive element B in A. Let ϕ0 be a positive normal linear functional on A . Let S be a semigroup and, {T(s) : s ε S} an antirepresentation of S as normal positive linear contraction operators on A . We find in this thesis equivalent conditions for the existence of a positive normal linear functional ϕ on A which is equivalent to ϕ0 and invariant under the semigroup {T(s) : s ε S} (i.e. ϕ(T(s)B) = ϕ(B) for all B in A and s ε S ). We also extend the concept of weakly-wandering sets, which was first introduced by Hajian-Kakutani, to weakly-wandering projections in A. We give a relation between the non-existence of weakly-wandering projections in A and the existence of positive normal linear functionals on A, invariant with respect to an antirepresentation {T(s) : s ε S} of normal *-homomorphisms on A . Finally we investigate the existence of a complete set of positive normal linear functionals on A which are invariant under the semigroup {T(s) : s ε S}. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis is a study of several spaces of continuous linear functionals on various function spaces with a natural norm inherited from a larger Banach space. The completeness of these normed linear spaces is studied in detail and several necessary and sufficient conditions are obtained in this regard. Since spaces of continuous linear functionals are inherently related to spaces of measures, their measure-theoretic counterparts are also studied. By using these counterparts, several necessary and sufficient conditions are obtained on the separability of these spaces of continuous linear functionals. / Ph. D.

Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

Because of several advantages, e.g. compact structure, high air-gap flux density, and high torque capability, the PM synchronous motor plays an important role in recent years. The basic principle of controlling a PMSM is based on vector control. The control performance is influenced by factors as the plant parameter variations, the external load disturbances, and the unmodeled or nonlinear dynamics. In the thesis, we apply a recently proposed robust 2DOF configuration to designing controllers for PMSM to achieve the robust asymptotical tracking under perturbations in both the motor and the controllers. Two design methods are adopted to implement the desired controllers, i.e. the linear algebraic method and the design method. The effect of the well-known internal model principle is addressed in the former design method. The merit of the latter design method is that both time and frequency domain design specifications can be easily included in the design procedure. Computer simulation results are displayed to illustrate the advantages of our designs.