  We show that the quadratic matrix equation $VW + \eta (W)W = I$, for given
$V$ with positive real part and given analytic mapping $\eta$ with some
positivity preserving properties, has exactly one solution $W$ with positive
real part. We point out the relevance of this result in the context of
operator-valued free probability theory and for the determination of the
asymptotic eigenvalue distribution of band or block random matrices. We also
address the problem of a numerical determination of the solution.
