Fredholm Theory with Applications to Random Operators
Title Granting Institution
Technische Universität Hamburg
Place of Title Granting Institution
This thesis is concerned with the Fredholm theory of bounded linear operators acting on Banach space valued sequence spaces. As an application, random operators are considered and studied in detail. One of the most important tools in the study of operators on sequence spaces is the concept of limit operators. The correspondence between an operator and its limit operators is studied regarding properties like spectrum, pseudospectrum and numerical range. It turns out that similar theorems can be formulated for all these properties, respectively. These results prove to be particularly useful in the case of random operators. Special attention is directed to the so-called Feinberg-Zee random hopping matrix, which, despite its simple appearance, seems to have a very complicated spectrum. With the help of new methods, improved upper and lower bounds to the spectrum are obtained. One of these lower bounds is an infinite sequence of Julia sets, which emphasizes the complexity of the spectrum of this particular operator.