Refined Heinz-Kato-L\"owner 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}$andarbitrary$X \in $\mathbb{R}$^{n \times n}$ $$\mathrm{\parallel }AXB\mathrm{\parallel }\le \mathrm{\parallel }{A}^{2}X{\mathrm{\parallel }}^{\frac{1}{2}}\mathrm{\parallel }X{B}^2{\mathrm{\parallel }}^{\frac{1}{2}}\mathrm{.}$$ 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|$,thecaseofequalityrequiresthat$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 $$\mathrm{\parallel }AXB\mathrm{\parallel }\le (1-c_{n,d})\mathrm{\parallel }{A}^{2}X{\mathrm{\parallel }}^{\frac{1}{2}}\mathrm{\parallel }X{B}^2{\mathrm{\parallel }}^{\frac{1}{2}}$$ 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.