It is common knowledge that if I have an operator $L$ defined on a Hilbert space $X$, then a good way to learn something about it's spectrum is to see if it is self-adjoint. If it is self-adjoint on $X$, then the spectrum is on the real axis. Alternatively if the operator $L$ is skew-adjoint on $X$ then the spectrum is on the imaginary axis. These two results are very nice, but of course (skew-) self-adjointness is not a necessary condition for $L$ to have (imaginary) real eigenvalues. Is there a theorem out there that says anything about the converse? e.g. If the operator $L$ has properties $a,b,$ and $c$, then it has no real eigenvalues. Or, if it has properties $d, e,$ and $f$, then it has no imaginary eigenvalues. Does anyone know about such a theorem? Of course the most general theorem (for linear operators) would be nice, but I am mostly concerned about closed operators and would also settle on a result for bounded (or even finite-dimensional!) operators.
I found this question a while back, but it seems to have generated little interest/response back then!