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Suppose $C=A+B$, where $A$ and $B$ are linear operators defined on an infinite dimensional Hilbert space with real spectra, but they are not necessarily self adjoint operators. Is it true that $C$ always have real spectrum? Any comment or reference will be greatly appreciated.

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  • $\begingroup$ Isn't an operator automatically self-adjoint if its spectrum is real? (Modulo some regularity assumption, of course). $\endgroup$ – Giuseppe Negro Oct 12 '18 at 16:26
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    $\begingroup$ A and B can be mapped into self-adjoint operators through similarity transformations : a = S A S^{-1}, b = T B T^{-1}, where a, b are self-adjoint operators. The question is whether there exists an invertible operator U where U (A+B) U^{-1} is self-adjoint. $\endgroup$ – ssongnul1 Oct 12 '18 at 16:31
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That's already false in dimension 2. $$\left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right)=\left( \begin{array}{cc} 4 & -1 \\ 1 & 1 \end{array}\right)+\left( \begin{array}{cc} -3 & 0 \\ 0 & 0\end{array} \right) .$$

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