  In this paper we discuss some spectral invariance results for non-smooth
pseudodifferential operators with coefficients in H\"older spaces. In analogy
to the proof in the smooth case of Beals and Ueberberg, 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 H\"older spaces,
we improve the characterization of non-smooth pseudodifferential operators in a
previous work by the authors.
