  The spectral theory of the Karoubi operator due to Cuntz and Quillen is
extended to general mixed (duchain) complexes, that is, chain complexes which
are simultaneously cochain complexes. Connes' coboundary map B can be viewed as
a perturbation of the noncommutative De Rham differential d by a polynomial in
the Karoubi operator. The homological impact of such perturbations is expressed
in terms of two short exact sequences.
