Define matrix weighted norm as $\|P\|_{AB}$ as $\|A^{1/2}PB^{-1/2}\|_{2}$, where $A$ and $B$ are positive definite matrices. Can I get some relationship between
\begin{equation} \|P\|_{AA}\|Q\|_{BB} \qquad \text{and} \qquad \|P\|_{BA}\|Q\|_{AB} \end{equation}
When $A=B$, then $\|P\|_{AA}\|Q\|_{BB} = \|P\|_{BA}\|Q\|_{AB}$. How about when $A$ and $B$ are not the same?