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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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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. 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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J. William Helton, Reza Rashidi Far, Roland Speicher. 2007-03-17. Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints. https://arxiv.org/abs/math/0703510

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