Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 527-545.

On the matrix range of random matrices

Authors: Malte Gerhold (1), Orr Moshe Shalit (2)
Author institution: (1) Institut fuer Mathematik und Informatik, Universitaet Greifswald, Walther-Rathenau-Strasse 47, 17487 Greifswald, Germany
(2) Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003, Israel

Summary: This note treats a simple minded question: \textit{what does a typical random matrix range look like?} We study the relationship between various modes of convergence for tuples of operators on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric on the other. In particular, we show that the matrix range of a tuple generating a continuous field of $C^*$-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.

DOI: http://dx.doi.org/10.7900/jot.2019dec04.2277
Keywords: matrix range, random matrices, matrix convexity

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