Norm inequalities for the spectral spread of Hermitian operators

TL;DR

This paper introduces a new measure called spectral spread for Hermitian operators and establishes submajorization inequalities relating to spectral dispersion, connecting various classical inequalities and norm bounds.

Contribution

The paper proposes the spectral spread as a novel measure of spectral dispersion and derives new submajorization inequalities involving this measure for Hermitian operators.

Findings

Established submajorization inequalities involving spectral spread.

Connected spectral spread inequalities to Tao's and Kittaneh's inequalities.

Derived norm inequalities for compact Hermitian operators.

Abstract

In this work we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the Arithmetic-Geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).