Stability of Cauchy-Stieltjes Kernel Families by Free and Boolean Convolutions Product

Let F(nu(j)) = {Q(mj)(nu j), m(j) is an element of(m(-)(nu j), m(+)(nu j))}, j=1,2, be two Cauchy-Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures nu 1 and nu 2. Introduce the set of measures F = F(nu(1)) boxed plus F(nu 2) = {Q(m1)(nu 1) boxed plus Q(m2)(nu 2), m(1) is an element of(m(-)(nu 1), m(+)(nu 1)) and m 2 is an element of(m(-)(nu 2), m(+)(nu 2))}. We show that if F remains a CSK family, (i.e., F = F(mu) where mu is a non-degenerate compactly supported measure), then the measures mu, nu(1) and nu(2) are of the Marchenko-Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution boxed plus by the additive Boolean convolution (sic). The cases where the additive free convolution boxed plus is replaced (in the definition of F) by the multiplicative free convolution boxed times or the multiplicative Boolean convolution boxed times are also studied.