Hanner's Inequality For Positive Semidefinite Matrices

TL;DR

This paper establishes a matrix version of Hanner's inequality for positive semidefinite matrices, extending known results across the entire range of p, and characterizes the equality condition, linking matrix norms to classical L^p spaces.

Contribution

It proves a new Hanner's inequality for positive semidefinite matrices valid for all p in [1,2], including the case p=2, and characterizes when equality holds.

Findings

The inequality holds for 1 ≤ p ≤ 2 and reverses for p ≥ 2.

Equality occurs if and only if Y is a scalar multiple of X.

The unit ball in the cone of positive matrices shares smoothness and convexity properties with L^p spaces.

Abstract

We prove an analogous Hanner's Inequality of $L^p$ spaces for positive semidefinite matrices. Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$. We show that the inequality $||X+Y||_p^p+||X-Y||_p^p\geq (||X||_p+||Y||_p)^p+(|||X||_p-||Y||_p|)^p$ holds for $1\leq p\leq 2$ and reverses for $p\geq 2$ when $X,Y\in M_{n\times n}(\mathbb{C})^+$. This was previously known in the $1<p\leq 4/3$, $p=2$, and $p\geq 4$ cases, or with additional special assumptions. We outline these previous methods, and comment on their failure to extend to the general case. We further show that there is equality if and only if $Y=cX$, which is analogous to the equality case in $L^p$. With the general inequality, it is confirmed that the unit ball in $C^{p}_+$ has the same moduli of smoothness and convexity as the unit ball in $L^p$.