Authors’ abstract: “We show that the quadratic matrix equation \(VW +\eta(W)W = I\), for given \(V\) with positive real part and given analytic mapping \(\eta\) 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 main theorem of this paper is proved by applying Banach’s fixed point theorem. The paper is well written.