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

  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},
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… 

Figures from this paper

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane
Let S be a positivity‐preserving symmetric linear operator acting on bounded functions. The nonlinear equation −1m=z+Sm with a parameter z in the complex upper half‐plane ℍ has a unique solution m
Operator-Valued Matrices with Free or Exchangeable Entries
We study matrices whose entries are free or exchangeable noncommutative elements in some tracial $W^*$-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and
Operator-Valued and Multivariate Free Berry-Esseen Theorems
We address the question of a Berry-Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued
Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory
We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an
Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem
We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series
Inhomogeneous Circular Law for Correlated Matrices
Quadratic Vector Equations On Complex Upper Half-Plane
We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on
Convergence of the largest singular value of a polynomial in independent Wigner matrices
For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth
Randomly coupled differential equations with elliptic correlations
We consider the long time asymptotic behavior of a large system of $N$ linear differential equations with random coefficients. We allow for general elliptic correlation structures among the
On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws
We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions,


Fixed points of holomorphic mappings for domains in Banach spaces
We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the
Extremal Problems of Interpolation Theory
We consider problems where one seeks m×m matrix valued H∞ functions w(ξ) which satisfy interpolation constraints and a bound (0.1) w∗(ξ)w(ξ) ≤ ρmin, |ξ| 0 where matrices X,R,C are N ×N matrices. When
Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory
Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory
A new application of random matrices: Ext(C^*_{red}(F_2)) is not a group
In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have
On the spectrum of random matrices
A study is made of the dis tr ibut ion of eigenvalues in a ce r ta in ensemble of random par t i c les that contains as a special case the ensemble used by Wlgner to give a s ta t i s t ica l descr
Lectures on the Combinatorics of Free Probability
Part I. Basic Concepts: 1. Non-commutative probability spaces and distributions 2. A case study of non-normal distribution 3. C*-probability spaces 4. Non-commutative joint distributions 5.
The Carathéodory and Kobayashi Metrics and Applications in Complex Analysis
The purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications.
A xed point theorem for holomorphic mappings
A combination reloading tray and die box for securely storing precision tools used in a process for reloading spent shells that includes a container having a hinged and latchable top cover. Shell