Definition:Symmetric 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.

We say that $\struct {\map D T, T}$ is symmetric if and only if:

$\innerprod {T x} y = \innerprod x {T y}$ for all $x, y \in \map D T$.

Also see

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

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.3$: Closed Operators and the Closure of Symmetric Operators