Have a look at the following excerpt of Tosio Kato (taken from Zeidler Applied functional analysis vol. I):
The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be self-adjoint which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, there exist a complete set (discrete or continuous) of eigenfunctions.
What does he mean? To me a self-adjoint operator $(A, D(A))$ on a Hilbert space $\mathcal{H}$ is a linear operator s.t. $A=A^\star$, which is equivalent to say that it is expressible in terms of a unique projection-valued measure $P_A$:
$A=\int_{-\infty}^\infty \lambda\, dP_A(\lambda).$
This is the best thing I can think of to match what Kato refers to. This is kind of incomplete, though. Where are those eigenfunctions Kato mentions? Also, a version of the spectral theorem holds for normal operators too. Why are they ruled out?