Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints

Abstract

We show that the quadratic matrix equation VW + η(W)W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs. This work bears on operator-valued free probability theory, in particular on the determination of the asymptotic eigenvalue distribution of band or block random matrices. © The Author 2007.