I've got this from Hiai and Kosaki paper, could someone explain to me how to derive $H^{1/2}XH^{1/2}=\left(\frac{2\lambda_i^{1/2}\lambda_j^{1/2}}{\lambda_i +\lambda_j} \right)\circ\left( \frac{1}{2}(HX+XH) \right)$ (red-boxed)
Also, what the meaning of "thanks to the standard $2\times 2$ matrix trick, one can reduce a proof to the case $H=K.$" What trick is this?
Any help would be appreciated.
Thanks so much
The trick is probably $$ \begin{bmatrix} H&0\\0&K\end{bmatrix}^{1/2} \begin{bmatrix} 0&X\\0&0\end{bmatrix} \begin{bmatrix} H&0\\0&K\end{bmatrix}^{1/2}, $$ since its norm is $\|H^{1/2}XK^{1/2}\|$ and $$ \begin{bmatrix} H&0\\0&K\end{bmatrix} \begin{bmatrix} 0&X\\0&0\end{bmatrix}+ \begin{bmatrix} 0&X\\0&0\end{bmatrix}\begin{bmatrix} H&0\\0&K\end{bmatrix} =\begin{bmatrix}0&HX+XK\\0&0\end{bmatrix}. $$ As for the fomula, doing the $2\times 2$ case to avoid dots, \begin{align} H^{1/2}XH^{1/2} &=\begin{bmatrix} \lambda_1^{1/2}&0\\0&\lambda_2^{1/2}\end{bmatrix} \begin{bmatrix} x&y\\ z&w\end{bmatrix} \begin{bmatrix} \lambda_1^{1/2}&0\\0&\lambda_2^{1/2}\end{bmatrix} =\begin{bmatrix} \lambda_1 x&\lambda_1^{1/2}\lambda_2^{1/2}y\\\lambda_1^{1/2}\lambda_2^{1/2} z&\lambda_2 w\end{bmatrix}\\[0.3cm] &=\begin{bmatrix} \lambda_1& \lambda_1^{1/2}\lambda_2^{1/2}\\ \lambda_1^{1/2}\lambda_2^{1/2}&\lambda_2\end{bmatrix} \circ X, \end{align}
and \begin{align} XH+HX&=\begin{bmatrix} \lambda_1 x&\lambda_1 y\\\lambda_2 z&\lambda_2w\end{bmatrix} +\begin{bmatrix} \lambda_1 x&\lambda_2 y\\\lambda_1 z&\lambda_2w\end{bmatrix} =\begin{bmatrix} 2\lambda_1 x&(\lambda_1+\lambda_2) y\\(\lambda_1+\lambda_2 )z&2\lambda_2w\end{bmatrix}\\[0.3cm] &=\begin{bmatrix} 2\lambda_1&\lambda_1+\lambda_2\\\lambda_1+\lambda_2&2\lambda_2\end{bmatrix}\circ X. \end{align} So, denoting the Hadamard inverse of $A$ by $A^\#$, \begin{align} H^{1/2}XH^{1/2} &=\begin{bmatrix} \lambda_1& \lambda_1^{1/2}\lambda_2^{1/2}\\ \lambda_1^{1/2}\lambda_2^{1/2}&\lambda_2\end{bmatrix}\circ \begin{bmatrix}2\lambda_1&\lambda_1+\lambda_2\\\lambda_1+\lambda_2&2\lambda_2\end{bmatrix}^\#\circ (HX+XH)\\[0.3cm] &=\begin{bmatrix} \lambda_1^{1/2}\lambda_1^{1/2}& \lambda_1^{1/2}\lambda_2^{1/2}\\ \lambda_1^{1/2}\lambda_2^{1/2}&\lambda_2^{1/2}\lambda_2^{1/2}\end{bmatrix}\circ \begin{bmatrix}\lambda_1+\lambda_1&\lambda_1+\lambda_2\\\lambda_1+\lambda_2&\lambda_2+\lambda_2\end{bmatrix}^\#\circ (HX+XH)\\[0.3cm] &=\begin{bmatrix} \frac{\lambda_1^{1/2}\lambda_1^{1/2}}{\lambda_1+\lambda_1}& \frac{\lambda_1^{1/2}\lambda_2^{1/2}}{\lambda_1+\lambda_2}\\ \frac{\lambda_1^{1/2}\lambda_2^{1/2}}{\lambda_1+\lambda_2}&\frac{\lambda_2^{1/2}\lambda_2^{1/2}}{\lambda_2+\lambda_2}\end{bmatrix}\circ (HX+XH) \end{align}