Refined Heinz–Kato–Loewner inequalities

Abstract

A version of the Cauchy–Schwarz inequality in operator theory is the following: for any two symmetric, positive definite matrices $A,B\in {\mathbb{R}}^{n\times n}$ and arbitrary $X\in {\mathbb{R}}^{n\times n}$

This inequality is classical and equivalent to the celebrated Heinz–Löwner, Heinz–Kato and Cordes inequalities. We characterize cases of equality: in particular, after factoring out the symmetry coming from multiplication with scalars $\Vert A^{2}X\Vert =1=\Vert X{B}^2\Vert$, the case of equality requires that $A$ and $B$ have a common eigenvalue ${\lambda }_i={\mu }_j$. We also derive improved estimates and show that if either ${\lambda }_i{\lambda }_j={\mu }_k^2$ or ${\lambda }_i^2={\mu }_j{\mu }_k$ does not have a solution, i.e. if $d>0$ where

then there is an improved inequality

for some $c_{n,d}>0$ that only depends only on $n$ and $d$. We obtain similar results for the McIntosh inequality and the Cordes inequality and expect the method to have many further applications.