The Hadamard product and the free convolutions

It is shown that if a probability measure nu is supported on a closed subset of (0, infinity), that is, its support is bounded away from zero, then the free multiplicative convolution of nu and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained. (C) 2017 Elsevier B.V. All rights reserved.