The atoms of the free additive convolution of two operator-valued distributions

2019/03/20 by Serban Belinschi, Serban T. Belinschi, Belinschi, Serban +4
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.FA #math.OA #math.PR

paper · pdf · doi:10.48550/arxiv.1903.09002

Second version, major modifications. Comments most welcome

openalex publication_date 2019/03/20 · openalex created_date 2019/04/01 · arxiv created 2020/05/15 · arxiv updated 2020/05/18 · openalex updated_date 2026/07/28

Abstract

Suppose that X_1 and X_2 are two selfadjoint random variables that are freely independent over an operator algebra B. We describe the possible operator atoms of the distribution of X_1+X_2 and, using linearization, we determine the possible eigenvalues of an arbitrary polynomial p(X_1,X_2) in case B=ℂ.

Citations

• Noncommutative hyperbolic metrics
• The free field: zero divisors, Atiyah property and realizations via unbounded operators
• Hölder Continuity of Cumulative Distribution Functions for Noncommutative Polynomials under Finite Free Fisher Information
• Invariant projections for operators that are free over the diagonal

Related

• Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem
• Universality of free random variables: atoms for non-commutative rational functions
• Analytic subordination for bi-free convolution
• Finite free convolutions of polynomials
• Operator-valued Jacobi parameters and examples of operator-valued distributions