Two selfadjoint operators T and S on a Hilbert space such that sp(ST) is not a subset of R.

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Consider

S = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , T = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

belonging to B(C^{2}). Then S and T are Hermetian, but sp(ST) = {i, −i} which is not a subset of R.