Usually we are told about perturbation theory for either Hermitian or for Symmetric operators (real or complex ones). However, I have encountered a problem, where whatever kind of representation of the total operator (an effective Hamiltonian of the system) in an "unperturbed + perturbation" form I choose: $\hat H = \hat H_0 + \hat V$ , I could not force both $\hat H_0$, and $\hat V$ be either Hermitian or Symmetric. Usually, most interesting problems in QM are such.
So, I wonder, is there the most general perturbation theory for this case? And what about the case when $\hat H_0$ has degenerate eigenvalues?
My particular case is the following: $\hat H_0$ is symmetric, and $\hat V$ can be made anti-symmetric. This is the best I can get.
An additional question: when $\hat V$, for instance, is not even a normal matrix, therefore, $\hat V^{\dagger} \hat V \ne \hat V \hat V^{\dagger}$) or even both $\hat H_0, \hat V$ are not diagonalizable (have Jordan blocks), is there a kind of perturbation theory for this case?