I learnt that the observables are self-adjoint operators working on wave functions which live in a Hilbert space. The eigenvalues of these operators are real and appear as outcome of measurements. Which eigenvalue comes as output, the probability is given by the weights of the orthonormal expansion of wavefunction on a basis composing the eigenvectors of the operator. And the final state would be the eigenvector corresponding to the eigenvalue emitted.
My question is: I'd like to know the importance of anti-self adjoint operators in quantum mechanics. $$\langle \psi,L\phi \rangle = - \langle L\psi,\phi \rangle$$. Their eigenvalues are imaginary. Do they make any sense to the theory of quantum mechanics?