Norm Inequalities in Operator Ideals

TL;DR

This paper introduces a novel technique for proving norm inequalities in operator ideals with unitarily invariant norms, extending known inequalities and deriving new ones, including applications to noncommutative $L^p$-spaces.

Contribution

A new method for establishing norm inequalities in operator ideals, applicable to a broad class of inequalities and extending results to noncommutative $L^p$-spaces.

Findings

Proved the L"owner-Heinz inequality using the new technique.

Derived several new inequalities by specialization.

Extended matrix inequalities to noncommutative $L^p$-spaces.

Abstract

In this paper we introduce a new technique for proving norm inequalities in operator ideals with an unitarily invariant norm. Among the well known inequalities which can be proved with this technique are the L\"owner-Heinz inequality, inequalities relating various operator means and the Corach-Porta-Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in $C^*$-algebras, in particular to the noncommutative $L^p$-spaces of a semi-finite von Neumann algebra.