Operadores universales y subespacios invariantes
The Invariant Subspace Problem is one of the most studied problems on Operator Theory in the last decades. In fact, it is still open in the Hilbert space setting. The purpose of this work is to present the most classic results concerning this problem and to study the approach based on universal oper...
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| Tipo de recurso: | tesis de maestría |
| Estado: | Versión publicada |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/93639 |
| Acceso en línea: | https://hdl.handle.net/11441/93639 |
| Access Level: | acceso abierto |
| Palabra clave: | Operadores, Teoría de Subespacios |
| Sumario: | The Invariant Subspace Problem is one of the most studied problems on Operator Theory in the last decades. In fact, it is still open in the Hilbert space setting. The purpose of this work is to present the most classic results concerning this problem and to study the approach based on universal operators. The text is organized as follows: In the first chapter we introduce the theory Banach algebras, focusing on spectral theory and Gelfand transform, two tools that will be fundamental in the development of the text. In the second chapter we provide a classical view of the invariant subspace problem in Hilbert spaces. We show two of the most important results on the existence of hyperinvariant subspaces: Lomonosov theorem and spectral theorem for normal operators. In the third chapter we study the tools to calculate invariant subspaces lattices for some classical operators, emphasizing on the need to use models to characterize these lattices. In chapter four we introduce the universal operators and we prove that the characterization of the invariant subspaces lattice of the hyperbolic automorphism composition operator in H2 would solve the invariant subspace problem. Finally, in chapter five we present the closest result to date: the characterization of the lattice of the parabolic nonautomorphism composition operator in H2. |
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