Refined Heinz-Kato-Löwner 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 $X\in {\mathbb{R}}^{n\times n}$ $$\Vert AXB\Vert \le \Vert A^{2}X{\Vert }^{\frac{1}{2}}\Vert X{B}^2{\Vert }^{\frac{1}{2}}.$$ 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 \begin{align*} d &= \min_{1 \leq i,j,k \leq n} \{ | \log{ λ_i} + \log{ λ_j} - 2\log{ μ_k}|:λ_i, λ_j \in σ(A), μ_k \in σ(B) \} &+\min_{1 \leq i,j,k \leq n}\{ | 2\log{λ_i} - \log{ μ_j} - \log{μ_k } |:λ_i \in σ(A), μ_j, μ_k \in σ(B) \}, \end{align*} then there is an improved inequality $$\Vert AXB\Vert \le (1-c_{n,d})\Vert A^{2}X{\Vert }^{\frac{1}{2}}\Vert X{B}^2{\Vert }^{\frac{1}{2}}$$ 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.