Generality of Lieb's Concavity Theorem

Abstract

We show that Lieb's concavity theorem holds more generally for any unitarily invariant matrix function $\varphi :{\mathbf{H}}_{+}^n\to \mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B)\mapsto \varphi [(B^{\frac{qs}{2}}{K}^{\ast }{A}^{ps}K{B}^{\frac{qs}{2}}{)}^{\frac{1}{s}}]$ on ${\mathbf{H}}_{+}^m\times {\mathbf{H}}_{+}^n$, for any $K\in {\mathbb{C}}^{m\times n},s\in (0,1],p,q\in [0,1],p+q\le 1$.

13 pages. arXiv admin note: text overlap with arXiv:1905.02194

References in corpus (1)

• Improvement on a Generalized Lieb's Concavity Theorem