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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 $B$?

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    Looks awfully like a generalized Sylvester equation...2011-08-09
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    If the equation $(a_{ij}b_{kl}-a_{kl}b_{ij})^2=0$ holds for all $i,j,k,l$, and the matrix entries are real or complex numbers, does it not follow that $B$ and $A$ linearly dependent?2011-08-09
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    @JyrkiLahtonen: please disregard my 2x2 case, that was calculated when I omitted the $\frac12$ and assumed $L\neq1$. But maybe the answer still boils down to linear dependence2011-08-09

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