Improvement on a Generalized Lieb's Concavity Theorem

TL;DR

This paper extends Lieb's concavity theorem to a broader class of unitary invariant matrix functions that are concave and satisfy H"older's inequality, demonstrating joint concavity under more general conditions.

Contribution

It generalizes Lieb's concavity theorem to any unitary invariant matrix function meeting specific concavity and inequality conditions, broadening its applicability.

Findings

Proves joint concavity of a generalized matrix function.

Extends previous results to a wider class of functions.

Improves upon recent work by Huang.

Abstract

We show that Lieb's concavity theorem holds more generally for any unitary invariant matrix function $\phi:\mathbf{H}_+^n\rightarrow \mathbb{R}_+^n$ that is concave and satisfies H\"older's inequality. Concretely, we prove the joint concavity of the function $(A,B) \mapsto\phi\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big] $ on $\mathbf{H}_+^n\times\mathbf{H}_+^m$, for any $K\in \mathbb{C}^{n\times m}$ and any $s,p,q\in(0,1], p+q\leq 1$. This result improves a recent work by Huang for a more specific class of $\phi$.