Refined Heinz-Kato-L\"owner inequalities

TL;DR

This paper refines classical operator inequalities by characterizing equality cases and providing improved bounds based on spectral properties of positive definite matrices, with potential broad applications.

Contribution

It characterizes equality cases in Heinz-Kato-L"owner inequalities and derives improved estimates depending on spectral data.

Findings

Characterization of equality cases involving common eigenvalues.

Derivation of improved inequalities with spectral gap conditions.

Extension of results to McIntosh and Cordes 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 $X \in \mathbb{R}^{n \times n}$ $$ \|AXB\| \leq \|A^2 X\|^{\frac{1}{2}} \|X B^2\|^{\frac{1}{2}}.$$ This inequality is classical and equivalent to the celebrated Heinz-L\"owner, Heinz-Kato and Cordes inequalities. We characterize cases of equality: in particular, after factoring out the symmetry coming from multiplication with scalars $ \|A^2 X\| = 1 = \|X B^2\|$, 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{ \lambda_i} + \log{ \lambda_j} - 2\log{ \mu_k}|:\lambda_i, \lambda_j \in \sigma(A), \mu_k \in \sigma(B) \} &+\min_{1 \leq i,j,k \leq n}\{ | 2\log{\lambda_i} - \log{ \mu_j} - \log{\mu_k } |:\lambda_i \in \sigma(A), \mu_j, \mu_k \in \sigma(B) \}, \end{align*} then there is an improved inequality $$ \|AXB\| \leq (1 - c_{n,d})\|A^2 X\|^{\frac{1}{2}} \|X B^2\|^{\frac{1}{2}}$$ 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.