Some inequalities for central moments of matrices

TL;DR

This paper investigates inequalities for central moments of matrices, focusing on whether bounds from commutative cases hold in noncommutative settings, and provides applications to matrix spread estimates.

Contribution

It proves that commutative bounds are tight for the fourth central moment in noncommutative matrices and offers new lower bounds for matrix spread.

Findings

Commutative bounds are tight for the fourth central moment in noncommutative matrices.

Provides lower estimates for the spread of Hermitian and normal matrices.

Establishes conditions under which inequalities hold in noncommutative matrix analysis.

Abstract

In this paper we shall study noncommutative central moment inequalities with a main focus on whether the commutative bounds are tight in the noncommutative case, or not. We prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case. As an application, we shall present some lower estimates of the spread of Hermitian and normal matrices as well.