  In this paper we show the invariance of the Fredholm index of non-smooth
pseudodifferential operators with coefficients in H\"older spaces. By means of
this invariance we improve previous spectral invariance results for non-smooth
pseudodifferential operators $P$ with coefficients in H\"older 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.
