Boolean Cumulants and Subordination in Free Probability

Abstract: We study subordination of free convolutions. We prove that for free random variables $X,Y$ and a Borel function $f$ the conditional expectation $E_\varphi\left[ (z-X-f(X)Yf^*(X))^{-1}| X\right]$, is a resolvent again. This result allows explicit calculation of the distribution of $X+f(X)Yf^*(X)$. The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.