Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Δημήτριος Χελιώτης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Το μέτρο Brown για μη κανονικούς τελεστές και η οριακή θεωρία τυχαίων πινάκων
Translated title:
The Brown measure for non-normal operators and the limit theory of random matrices
Summary:
We study the Brown measure of certain non–Hermitian operators arising from Voiculescu’s free probability theory. Usually those
operators appear as the limit in ⋆-moments of certain ensembles of non–Hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in ⋆-free variables. Other explicit examples include R–diagonal elements and elliptic elements, for which the Brown measure was already known, and a
new class of triangular–elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the Hermitian reduction method, as considered in the physical literature, combined with subordination and the linearization trick.
Main subject category:
Science
Keywords:
Brown measure, spectral measure, operators
