Definition:Composition 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:


 * $\map D {S T} = \set {x \in \map D T : T x \in \map D S}$

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


 * $\map {\paren {S T} } x = \map {\paren {S \circ T} } x$

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

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