Limit operators and applications on the space of essentially bounded functions [Elektronische Ressource] / vorgelegt von Marko Lindner

Limit Operators and Applicationson the Spaceof Essentially Bounded Functionsvon der Fakultat¨ fur¨ Mathematikder Technischen Universitat¨ ChemnitzgenehmigteDissertationzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)vorgelegt vonDiplom-Mathematiker Marko Lindner,geboren am 20. Oktober 1973 in Zschopau,eingereicht am 7. April 2003.Gutachter: Prof. Dr. B. SilbermannProf. Dr. S. N. Chandler-Wilde¨Prof. Dr. W. SprossigTag der Verteidigung: 8. Dezember 2003IntroductionThis thesis is concerned with the description of the behaviour at infinity of so called∞ ∞ nband-dominated operators on L :=L (R ). We are especially interested in a prop-erty called invertibility at infinity which turns out to be the key to some other inter-esting features of the operator under consideration, including Fredholmness and theapplicability of several approximation methods. We will discuss and prove all thesecross connections.Our main tool for the investigation of invertibility at infinity are limit operators.Forthepracticabilityofthederivedcriteriaitisessentialtohavesomegoodknowledgeof limit operators; and on the journey through this subject we prove some previouslyunknown facts which – by the way – are beautiful and interesting on their own. Thisis why parts of this thesis may be understood as an approach to the understanding oflimit operators – a process that certainly still needs some time.