If we consider the self-adjoint operator $L= L_0 + h$ on the appropriate Sobolev spaces of maps from $S^1$ to $\mathbb{R}^N$, say, where $L_0$ is a first order self-adjoint operator and $h$ is a symmetric matrix-valued 0th order term, what happens to the spectrum as we move continuously from $L=L_0 +h$ to $L'=L_0+h'$? Here $h'$ need no longer be symmetric so that $L'$ does not remain self-adjoint, but $h'$ must retain the same set of eigenvalues as $h$. In particular, does $L'$ remain hyperbolic (i.e. spectrum does not meet the imaginary axis)? I know there may well be a good deal of material on perturbations of self-adjoint operators, but I can't seem to find anything that helps me with the case in hand. Thanks.
