Subordination methods for free deconvolution

In this paper, we give subordination functions for free additive and free multiplicative deconvolutions in some domain of the complex half-plane, under the condition that the distributions admit moments, respectively, of second order for the additive deconvolution and of fourth order for the multiplicative one. Our method of proof allows us to give an algorithm to calculate these subordinations functions, and thus the associated Cauchy transforms, for complex numbers with imaginary part bigger than a parameter depending on the measure to deconvolve. This reduces the problem of free deconvolutions to the one of the classical deconvolution with a Cauchy distribution and thus combined with known methods for the latter problem we are able to solve the deconvolution problem for the scalar case. We present also an extension of these results to the case of operator valued distributions.