Definition:Composition of Densely-Defined Linear Operators
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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 $\struct {\map D T, T}$.
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $13.1$: Definitions