Non-commutative Hardy inequalities

TL;DR

This paper extends Hardy's inequality to positive semi-definite operators and trace functions, introduces the tracial geometric mean, and generalizes Carleman's inequality, broadening the scope of operator inequalities.

Contribution

It generalizes Hardy's inequality to operators for all p>1, introduces the tracial geometric mean, and extends classical inequalities to the operator setting.

Findings

Extended Hardy's inequality to positive semi-definite operators for 1<p<=2

Generalized Hardy's inequality to operators under trace for p>1

Introduced the tracial geometric mean and generalized Carleman's inequality

Abstract

We extend Hardy's inequality from sequences of non-negative numbers to sequences of positive semi-definite operators if the parameter p satisfies 1<p<=2, and to operators under a trace for arbitrary p>1. Applications to trace functions are given. We introduce the tracial geometric mean and generalize Carleman's inequality. Key words and phrases: Hardy's inequality, positive operator, trace function, geometric mean, Carleman's inequality.