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 if and only if:

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

Also see

• Results about self-adjoint densely-defined linear operators can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.1$: Adjoints of Unbounded Operators