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    Item type:Publication, Access status: with files ,
    Stable approximation and spectral theory
    (2024)
    Ukena, Riko
    ;
    Lindner, Marko
    This work presents methods for the approximation of solutions of linear systems involving band-dominated operators on l^p spaces. In particular, conditions for the applicability of the periodic finite section method are found and a method for the approximation of pseudospectra of band operators is derived. Additionally, Dirichlet eigenvalues of 1D discrete Schrödinger operators are studied. For integer-valued potential and a subclass of periodic potentials the absence of Dirichlet eigenvalues at energy 0 is proven. For Sturmian potentials, the spectral convergence for the periodic approximation and a method for a precise localization of Dirichlet eigenvalues are derived.
    Publicationtype: Doctoral Thesis
      41  162
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    Item type:Publication, Access status: with files ,
    Fredholm Theory with Applications to Random Operators
    (2016)
    Hagger, Raffael
    ;
    Lindner, Marko
    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.
    Publicationtype: Doctoral Thesis
      848  2530