Invariance of the Fredholm index and spectrum of non-smooth pseudodifferential operators

In this paper we show the invariance of the Fredholm index of non-smooth pseudodifferential operators with coefficients in Holder spaces. By means of this invariance we improve previous spectral invariance results for non-smooth pseudodifferential operators $P$ with coefficients in Holder spaces. For this purpose we approximate $P$ with smooth pseudodifferential operators and use a spectral invariance result of smooth pseudodifferential operators. Then we get the spectral invariance result in analogy to a proof of the spectral invariance result for non-smooth differential operators by Rabier.