Computing extreme singular values of free operators

TL;DR

This paper extends the computation of spectral edges from self-adjoint to non-self-adjoint free operators, providing variational formulas for extreme singular values in random matrix theory.

Contribution

It introduces variational formulas for the largest and smallest singular values of non-self-adjoint free operators, advancing spectral analysis methods.

Findings

Derived variational formulas for extreme singular values

Extended spectral edge computation to non-self-adjoint operators

Facilitated analysis of general random matrices via free operator spectra

Abstract

A recent development in random matrix theory, the intrinsic freeness principle, establishes that the spectrum of very general random matrices behaves as that of an associated free operator. This reduces the study of such random matrices to the deterministic problem of computing spectral statistics of the free operator. In the self-adjoint case, the spectral edges of the free operator can be computed exactly by means of a variational formula due to Lehner. In this note, we provide variational formulas for the largest and smallest singular values in the non-self-adjoint case.