Point Spectrum of Symmetric Densely-Defined Linear Operator is Real

Theorem
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\struct {\map D T, T}$ be a symmetric densely-defined linear operator.

Let $\map {\sigma_p} T$ be the point spectrum of $\struct {\map D T, T}$.

Then:


 * $\map {\sigma_p} T \subseteq \R$

Proof
If $\map {\sigma_p} T = \O$, the result is immediate.

Let $\lambda \in \map {\sigma_p} T$.

Then, from Point Spectrum of Densely-Defined Linear Operator consists of its Eigenvalues, there exists $x \in \HH \setminus \set 0$ such that:


 * $T x = \lambda x$

Then, we have:


 * $\innerprod {T x} x = \innerprod {\lambda x} x = \lambda \innerprod x x$

and:

So, we have:


 * $\paren {\lambda - \overline \lambda} \innerprod x x = 0$

Since $x \ne 0$, we have $\innerprod x x \ne 0$, so:


 * $\lambda = \overline \lambda$

From Complex Number equals Conjugate iff Wholly Real, we then have:


 * $\lambda \in \R$

So:


 * $\map {\sigma_p} T \subseteq \R$

from the definition of set inclusion.