Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems

The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above formalism to asymptotic hyperstability of dynamic systems are discussed later on.

First-principles study of the electronic structure and dynamical electronic excitations of strong spin-orbit metal Pb: bulk, surface and nanosized films

En la presente tesis se ha realizado el estudio de primeros principios (esto es, sinhacer uso de parámetros ajustables) de la estructura electrónica y la dinámica deexcitaciones electrónicas en plomo, tanto en volumen como en superficie y en formade películas de espesor nanométrico. Al presentar el plomo un número atómico alto(82), deben tenerse en cuenta los efectos relativistas. Con este fin, el doctorando haimplementado el acoplo espín-órbita en los códigos computacionales que hanrepresentado la principal herramienta de trabajo.En volumen, se han encontrado fuertes efectos relativistas asi como de lalocalización de los electrones, tanto en la respuesta dieléctrica (excitacioneselectrónicas colectivas) como en el tiempo de vida de electrones excitados. Lacomparación de nuestros resultados con medidas experimentales ha ayudado aprofundizar en dichos efectos.En el estudio de las películas a escala nanométrica se han hallado fuertes efectoscuánticos debido al confinamiento de los estados electrónicos. Dichos efectos semanifiestan tanto en el estado fundamental (en acuerdo con estudiosexperimentales), como en la respuesta dieléctrica a través de la aparición y dinámicade plasmones de diversas características. Los efectos relativistas, a pesar de no serimportantes en la estructura electrónica de las películas, son los responsables de ladesaparación del plasmón de baja energía en nuestros resultados.

Pumped double quantum dot with spin-orbit coupling

We study driven by an external electric field quantum orbital and spin dynamics of electron in a one-dimensional double quantum dot with spin-orbit coupling. Two types of external perturbation are considered: a periodic field at the Zeeman frequency and a single half-period pulse. Spin-orbit coupling leads to a nontrivial evolution in the spin and orbital channels and to a strongly spin-dependent probability density distribution. Both the interdot tunneling and the driven motion contribute into the spin evolution. These results can be important for the design of the spin manipulation schemes in semiconductor nanostructures.

Spin-related phenomena in magnetic and spin-orbit couppled bose-einstein condensates

129 p.

Spin-helical Dirac states in graphene induced by polar-substrate surfaces with giant spin-orbit interaction: a new platform for spintronics

Spintronics, or spin electronics, is aimed at efficient control and manipulation of spin degrees of freedom in electron systems. To comply with demands of nowaday spintronics, the studies of electron systems hosting giant spin-orbit-split electron states have become one of the most important problems providing us with a basis for desirable spintronics devices. In construction of such devices, it is also tempting to involve graphene, which has attracted great attention because of its unique and remarkable electronic properties and was recognized as a viable replacement for silicon in electronics. In this case, a challenging goal is to lift spin degeneracy of graphene Dirac states. Here, we propose a novel pathway to achieve this goal by means of coupling of graphene and polar-substrate surface states with giant Rashba-type spin-splitting. We theoretically demonstrate it by constructing the graphene@BiTeCl system, which appears to possess spin-helical graphene Dirac states caused by the strong interaction of Dirac and Rashba electrons. We anticipate that our findings will stimulate rapid growth in theoretical and experimental investigations of graphene Dirac states with real spin-momentum locking, which can revolutionize the graphene spintronics and become a reliable base for prospective spintronics applications.

Closed G2 forms and special metrics

170 p.