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Refined Heinz-Kato-Löwner inequalities

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 $λ_i = μ_j$. We also derive improved estimates and show that if either $λ_i λ_j = μ_k^2$ or $λ_i^2 = μ_j μ_k$ 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 $$ \|AXB\| \leq (1 - c_{n,d})\|A^2 X\|^{\frac{1}{2}} \|X B^2\|^{\frac{1}{2}}$$ 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.