Definition:Closed 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 $\struct {\HH \times \HH, \norm \cdot_{\HH \times \HH} }$ be the direct product $\HH \times \HH$ equipped with the direct product norm.

We say that $\struct {\map D T, T}$ is closed if:

$\set {\tuple {x, T x} \in \HH \times \HH : x \in \map D T}$

is a closed subset of $\HH \times \HH$.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.2$: Closed Operators and the Closure of Symmetric Operators