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Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints
J. William Helton, Reza Rashidi Far, Roland Speicher
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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. 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.
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