On the Araki-Lieb-Thirring inequality

TL;DR

This paper establishes a new lower bound complementing the Araki-Lieb-Thirring inequality for positive matrices, and extends the inequality to general matrices, enriching the theoretical framework of matrix inequalities.

Contribution

It introduces a novel lower bound for trace expressions involving positive matrices and generalizes the ALT inequality beyond positive matrices.

Findings

Derived a lower bound for trace expressions of positive matrices.

Extended the ALT inequality to general matrices.

The bounds involve matrix norms and differ from Kantorovich-type inequalities.

Abstract

We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices $A$ and $B$, by giving a lower bound on the quantity $\trace[A^r B^r A^r]^q$ in terms of $\trace[ABA]^{rq}$ for $0\le r\le 1$ and $q\ge0$, whereas the ALT inequality gives an upper bound. The bound contains certain norms of $A$ and $B$ as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (\textit{Math. Inequal. Appl.} \textbf{8}(2005) pp. 373--378) and others. Secondly, we also prove a generalisation of the ALT inequality to general matrices.