# 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?

For an example of the former, try e.g. $$\begin{pmatrix} 2 & - 1 \\ 0 & 1 \end{pmatrix},$$ whose second eigenvector is $(1,1)$.
Finally, you can have an non-diagonalizable operator with real eigenvalues. Its Jordan normal form must then contain non-trivial Jordan cells. An example would be $$\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix},$$ also known as the fermion creation operator $a^\dagger$ or as the momentum rasing operator $l_+$ in spin-1/2 representation.