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

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. We point out the relevance of this result in the context of operator-valued free probability theory and for the determination of the asymptotic eigenvalue distribution of band or block random matrices. We also address the problem of a numerical determination of the solution.

14 pages, 2 figures

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