Summary: The free convolution \(\boxplus\) is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a noncommutative probability space. In the same way, the rectangular free convolution \(\boxplus_{\lambda}\) allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices.
In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups \((\mu_t)\) of probability measures such that \(\mu _{0} = \delta _{0}\) and such that for all \(t > 0\) and all probability measures \(\nu\), \(\mu_t\boxplus\nu\) (or, in the rectangular context, \(\mu_t\boxplus_{\lambda}\nu\)) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for \(\boxplus\), we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for \(\mu \) in a \(\boxplus_{\lambda}\)-continuous semigroup, \(\mu\boxplus_{\lambda}\nu\) either has an atom at the origin or doesn’t put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of \(\mu\boxplus_{\lambda}\nu\) except on a negligible set of points, as well as existence and continuity of a density everywhere.