# Improvement on a Generalized Lieb's Concavity Theorem

@article{Huang2019ImprovementOA, title={Improvement on a Generalized Lieb's Concavity Theorem}, author={De Huang}, journal={arXiv: Functional Analysis}, year={2019} }

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 Holder'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…

## 2 Citations

### Generality of Lieb's Concavity Theorem

- Mathematics
- 2019

We show that Lieb's concavity theorem holds more generally for any unitarily invariant matrix function $\phi:\mathbf{H}^n_+\rightarrow \mathbb{R}$ that is monotone and concave. Concretely, we prove…

### FA ] 3 0 M ay 2 01 9 Generality of Lieb ’ s Concavity Theorem

- Mathematics
- 2019

We show that Lieb’s concavity theorem holds more generally for any unitarily invariant matrix function φ : H+ → R that is monotone and concave. Concretely, we prove the joint concavity of the…

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