Estudamos a álgebra de comparação do círculo, descrevemos o espaço de Gelfand do seu quociente pelos compactos e damos uma fórmula para o índice dos seus operadores de Fredholm. Depois generalizamos o resultado para as matrizes com elementos na álgebra de comparação e damos uma aplicação para operadores diferenciais no círculo. / We study the comparison algebra on the circle, we describe the Gelfand space of its quotient by the compacts and we give a formula to compute the index of its Fredholm operators. After that, we generalize the result to the matrices with entries in the comparison algebra and give an application to differential opperators in the circle.

Estudamos a álgebra de comparação do círculo, descrevemos o espaço de Gelfand do seu quociente pelos compactos e damos uma fórmula para o índice dos seus operadores de Fredholm. Depois generalizamos o resultado para as matrizes com elementos na álgebra de comparação e damos uma aplicação para operadores diferenciais no círculo. / We study the comparison algebra on the circle, we describe the Gelfand space of its quotient by the compacts and we give a formula to compute the index of its Fredholm operators. After that, we generalize the result to the matrices with entries in the comparison algebra and give an application to differential opperators in the circle.

Die abgeschlossenen Erweiterungen der sogenannten geometrischen Operatoren (Spin-Dirac, Gauß-Bonnet und Signatur-Operator) auf Mannigfaltigkeiten mit metrischen Hörnern sind Fredholm-Operatoren und ihr Index wurde von Matthias Lesch, Norbert Peyerimhoff und Jochen Brüning berechnet. Es wurde gezeigt, dass die Einschränkungen dieser drei Operatoren auf eine punktierte Umgebung des singulären Punkts unitär äquivalent zu irregulär singulären Operator-wertigen Differentialoperatoren erster Ordnung sind. Die Lösungsoperatoren der dazugehörigen Differentialgleichungen definierten eine Parametrix, mit deren Hilfe die Fredholmeigenschaft bewiesen wurde. In der vorliegenden Doktorarbeit wird eine Klasse von irregulären singulären Differentialoperatoren erster Ordnung, genannt Horn-Operatoren, eingeführt, die die obigen Beispiele verallgemeinern. Es wird bewiesen, dass ein elliptischer Differentialoperator erster Ordnung, dessen Einschränkung auf eine punktierte Umgebung des singulären Punkts unitär äquivalent zu einem Horn-Operator ist, Fredholm ist, und sein Index wird berechnet. Schließlich wird dieser abstrakte Index-Satz auf geometrische Operatoren auf Mannigfaltigkeiten mit "multiply warped product"-Singularitäten angewendet, welche eine wesentliche Verallgemeinerung der metrischen Hörner darstellen. / The closed extensions of geometric operators (Spin-Dirac, Gauss-Bonnet and Signature operator) on a manifold with metric horns are Fredholm operators, and their indices were computed by Matthias Lesch, Norbert Peyerimhoff and Jochen Brüning. It was shown that the restrictions of all three operators to a punctured neighbourhood of the singular point are unitary equivalent to a class of irregular singular operator-valued differential operators of first order. The solution operators of the corresponding differential equations defined a parametrix which was applied to prove the Fredholm property. In this thesis a class of irregular singular differential operators of first order - called horn operators - is introduced that extends the examples mentioned above. It is proved that an elliptic differential operator of first order whose restriction to the neighbourhood of the singular point is unitary equivalent to a horn operator is Fredholm and its index is computed. Finally, this abstract index theorem is applied to compute the indices of geometric operators on manifolds with multiply warped product singularities that extend the notion of metric horns considerably.