Year: 2019
Type: preprint
Abstract: We show that Lieb's concavity theorem holds more generally for any unitary invariant matrix function $ϕ:\mathbf{H}_+^n\rightarrow \mathbb{R}_+^n$ that is concave and satisfies Hölder's inequality. Concretely, we prove the joint concavity of the function $(A,B) \mapstoϕ\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 $ϕ$.
Source: arXiv (Cornell University)
Author: Huang, De
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2964216268