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Consider a matrix with two entries being some operator or matrix $$D=\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}.$$ I want to construct another $2\times2$ matrix $S$ such that $$SDS^{-1} = \begin{bmatrix} 0 & CAC^{-1} \\ C^{-1}BC & 0 \end{bmatrix}$$ where $C$ is some operator or matrix.

Is it possible or not?

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No, it is not possible. The matrices $D$ and $\begin{bmatrix} 0 & CAC^{-1} \\ C^{-1}BC & 0 \end{bmatrix}$ are not similar in general; they do not even have the same characteristic polynomial in general.

For a counterexample, set $A=\left( \begin{array}{cc} a & 0\\ 0 & b \end{array} \right) $ and $B=\left( \begin{array}{cc} c & 0\\ 0 & d \end{array} \right) $ and $C=\left( \begin{array}{cc} 2 & 1\\ 1 & 0 \end{array} \right) $.

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  • $\begingroup$ Thanks for the answer! Do you think it possible with any additional condition? Actually, I am thinking about the case when $A,B$ are simple functions of $x$ and $C=\exp(−\mathrm{i}a\frac{d}{dx})$ is a differential operator with constant a. $\endgroup$ – xiaohuamao Feb 4 at 23:31
  • $\begingroup$ @xiaohuamao: Your $C$ looks more like a shift operator than a differential operator :) But no, I'M afraid I don't know any conditions that could help. $\endgroup$ – darij grinberg Feb 4 at 23:37
  • $\begingroup$ Yes, it's the shift operator. Thanks for your opinion. I'll wait for a little while and accept your answer if no other. $\endgroup$ – xiaohuamao Feb 4 at 23:46

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