For $A,B,L$ linear operators, when is there a linear operator $X\{A,B\}$ such that
$$ALA^{-1}+BLB^{-1}=2 XLX^{-1}$$
can be solved independently for all $L$ only depending on $A$ and $B$?
For $A,B,L$ linear operators, when is there a linear operator $X\{A,B\}$ such that
$$ALA^{-1}+BLB^{-1}=2 XLX^{-1}$$
can be solved independently for all $L$ only depending on $A$ and $B$?