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- W2964161380 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–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{align*} d &= min_{1 leq i,j,k leq n} left{ left| log{ lambda_i} + log{ lambda_j} - 2log{ mu_k} right|:lambda_i, lambda_j in sigma(A), mu_k in sigma(B) right} &+min_{1 leq i,j,k leq n} left{ left| 2log{lambda_i} - log{ mu_j} - log{mu_k } right|:lambda_i in sigma(A), mu_j, mu_k in sigma(B) right}, 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." @default.
- W2964161380 created "2019-07-30" @default.
- W2964161380 creator A5054778975 @default.
- W2964161380 date "2018-10-22" @default.
- W2964161380 modified "2023-09-25" @default.
- W2964161380 title "Refined Heinz–Kato–Loewner inequalities" @default.
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- W2964161380 doi "https://doi.org/10.4171/jst/239" @default.
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