  We address the question of a Berry-Esseen type theorem for the speed of
convergence in a multivariate free central limit theorem. For this, we estimate
the difference between the operator-valued Cauchy transforms of the normalized
partial sums in an operator-valued free central limit theorem and the Cauchy
transform of the limiting operator-valued semicircular element. Since we have
to deal with in general non-self-adjoint operators, we introduce the notion of
matrix-valued resolvent sets and study the behavior of Cauchy transforms on
them.
