Spectal Invariance, Fredholmness of Pseudodifferential Operators and Applications

• Applicant: Dr. Christine Pfeuffer
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 442104339

The primary goal of this project is to improve the existing spectral invariance statements for non-smooth pseudodifferential operators. The main tool for proving this result was the characterization of these objects via mapping properties. We can possibly reach our aim by means of the improvement of this characterization. Another ansatz was to use the Fredhlom property of non-smooth pseudodifferential operators for a special subclass of those operators. We first established some conditions for the invariance of the Fredholm index within the framework of this project. Then we exploited this result in order to improve the existing spectral invariance statement. We intend to generalize the Fredhom property of so-called localization operators, a special subclass of non-smooth pseudodifferential operators, to the quasi-Banach space case. The approximation of pseudodifferential operators with localization operators may help us to show the Fredholm property for a more general class of pseudodifferential operators. Additionally we want to investigate some applications of the characterization of non-smooth pseudodifferential operators.

• DFG Programme: Research Grants