Let $A: D_A \rightarrow H$ be a symmetric linear operator defined on $D_A \subseteq H$. Define the deficiency index of $A$ at $z \in \mathbb{C} \setminus \mathbb{R}$ to be $\dim( \ker (A^* - \bar{z}))$. Then I want to show that the deficiency index is constant on the upper half-plane and constant on the lower half-plane.
The assertion is given in Lax, section 33.2, Theorem 3 (i), but he does not prove it.
If $A$ is also closed, then I am able to show that if the deficiency index is $0$ then it is so on the whole half plane, which is a consequence of the fact that the resolvent set is open. However, I don't know about the general case.
The proof is in Reed/Simon “Methods of Modern Mathematicsl Physics”, Vol. 2, Section X: Theorem X.1