Improvement on a Generalized Lieb's Concavity Theorem

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 $\phi$.