Show that (G_{A B}(t)=G_{B A}(t-i beta hbar)) and use the cyclic property of the traces to derive

Question:

Show that $G_{A B}(t)=G_{B A}(t-i \beta \hbar)$ and use the cyclic property of the traces to derive the fluctuation-dissipation theorem $\hat{\chi}_{A B}^{\prime \prime}(\omega)=\frac{1}{2 \hbar}\left(1-e^{-\beta \hbar \omega}\right) S_{A B}(\omega)$.

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