Definition:Self-Adjoint Densely-Defined Linear Operator

Definition
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\struct {\map D T, T}$ be a densely defined linear operator on $\HH$

Let $\struct {\map D {T^\ast}, T^\ast}$ be the adjoint of $\struct {\map D T, T}$.

We say that $\struct {\map D T, T}$ is self-adjoint :


 * $\struct {\map D {T^\ast}, T^\ast} = \struct {\map D T, T}$