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We develop an analytic theory of operator-valued additive free convolution interms of subordination functions. In contrast to earlier investigations ourfunctions are not just given by power series expansions, but are defined asFrechet analytic functions in all of the operator upper half plane.Furthermore, we do not have to assume that our state is tracial. Combining thisnew analytic theory of operator-valued free convolution with Anderson'sselfadjoint version of the linearization trick we are able to provide asolution to the following general random matrix problem: How can we calculatethe asymptotic eigenvalue distribution of a polynomial evaluated in independentrandom matrices with known asymptotic eigenvalue distributions?

Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem