Refined Heinz-Kato-Löwner inequalities (Q1733073)

Summary: 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-Löwner, 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 \[ \begin{aligned} d &= \underset{1 \leq i,j,k \leq n}{\min} \{ | \log \lambda_i + \log \lambda_j - 2\log \mu_k |:\lambda_i, \lambda_j \in \sigma(A), \mu_k \in \sigma(B)\} \\ &+\underset{1 \leq i,j,k \leq n}{\min} \{|2\log\lambda_i - \log \mu_j - \log\mu_k |:\lambda_i \in \sigma(A), \mu_j, \mu_k \in \sigma(B)\}, \end{aligned} \] 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.