Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

@article{Helton2007OperatorvaluedSE,
  title={Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints},
  author={J. William Helton and Reza Rashidi Far and Roland Speicher},
  journal={International Mathematics Research Notices},
  year={2007},
  volume={2007}
}
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… 

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