Is $\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}$ similar to $\begin{bmatrix} 0 & CAC^{-1} \\ C^{-1}BC & 0 \end{bmatrix}$ by some transformation?

Question

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?

Answer 1

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) $.