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 $\varphi :{\mathbf{H}}_{+}^n\to {\mathbb{R}}_{+}^n$ that is concave and satisfies Hölder's inequality. 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 ${\mathbf{H}}_{+}^n\times {\mathbf{H}}_{+}^m$, for any $K\in {\mathbb{C}}^{n\times m}$ and any $s,p,q\in (0,1],p+q\le 1$. This result improves a recent work by Huang for a more specific class of $\varphi$.

arXiv commentarXiv admin note: text overlap with arXiv:1904.03304

References in corpus (2)

• Generalizing Lieb's Concavity Theorem via Operator Interpolation
• A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices

Cited by in corpus (1)

• Generality of Lieb's Concavity Theorem