Spectral Invariance of Non-Smooth Pseudo-Differential Operators

We discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Holder spaces in this paper. In analogy to the proof in the smooth case of Beals and Ueberberg, c.f. (Duke Math J 44(1):45-57, 1977; Manuscripta Math 61(4):459-475, 1988), we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in Holder spaces, we improve the characterization of non-smooth pseudodifferential operators of A. and P., c.f. (Abels and Pfeuffer, Characterization of non-smooth pseudodifferential operators. arXiv:1512.01127,2015)