I was attempting a simpler version of this question mathoverflow.net/questions/204769/… , which no one has answered yet!

In particular I am interested in first order differential operators of the form $L=L_0+h$ where $L_0$ is first order self-adjoint, but the zeroth order term $h$ is not necessarily self-adjoint. Then yes, I am looking for the most general condition on $h$ that ensures everything in the spectrum has nonzero real part.

This just looks like the condition for an operator to not be skew-self-adjoint...

Alex, could you provide a link to the proof of this statement? In particular, I want a proof that applies to first order differential operators.