Definition:Sum of Densely-Defined Linear Operators

Definition

Let $\HH$ be a Hilbert space.

Let $\struct {\map D S, S}$ and $\struct {\map D T, T}$ be densely-defined linear operators.

Let:

$\map D {S + T} = \map D S \cap \map D T$

Define $S + T : \map D {S + T} \to \HH$ by:

$\map {\paren {S + T} } x = S x + T x$

for each $x \in \map D {S + T}$.

We say that $\struct {\map D {S + T}, S + T}$ is the sum of $\struct {\map D S, S}$ and $\struct {\map D T, T}$.