# Is there any non-hermitian operator on Hilbert Space with all real eigenvalues?

The property of hermitian is the sufficient condition for eigenvalue being real. Is there any non-hermitian operator on Hilbert Space with all real eigenvalues? If there exist, then can all eigenstates be orthogonal to each other? And these operators have any application in Quantum mechanics?

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For an example of the former, try e.g. $$\begin{pmatrix} 2 & - 1 \\ 0 & 1 \end{pmatrix},$$ whose second eigenvector is $(1,1)$.