An Operator Like Sturm-Liouville with Orthogonal Eigenfunctions

Question

Do we have any other operator like Sturm-liouville operator that give us real eigenvalues and orthogonal eigenfunctions? I know the Hermitian operators have this property but I like to see an example other than Sturm-Liouville. And what about non Hermitian ones?

Answer 1

I can't tell from your post if you know it already, but there exists the spectral theorem for normal operators:

Let $(H, \langle \cdot, \cdot \rangle)$ a Hilbert space and $T: H \to H$ a compact, normal operator. Then $T$ is of the form $$ Tx = \sum_{n} \lambda_n \langle x, e_n \rangle e_n \qquad \text{for all } x \in H$$ for a (maybe finite) sequence $(\lambda_n)_{n \in \mathbb N}$ in $\mathbb C \setminus \{0\}$ tending to $0$ and vectors $(e_n)_{n \in \mathbb N}$ in $H$ with

1. $T e_n = \lambda_n e_n$ for all $n \in \mathbb N$.
2. $\langle e_n, e_m \rangle = \delta_{n, m}$ for all $n, m \in \mathbb N$.

There are further interesting properties in the setting of the theorem. Also there are versions of the theorem with less assumptions to the operator (namely bounded instead of compact or even unbounded normal operators). I hope that was what you ask for :)