Plain-English statement
The unique right-generator value defines a genuine linear map on the generator domain.
Mathematical statement
L : D(L) -> M is linear and its value Lf is the norm limit of (P_h f - f)/h as h decreases to zero.
Intuition
Existence comes from domain membership, uniqueness removes the arbitrary witness, and quotient linearity makes the resulting operator linear.
Conditions
- a continuous-linear semigroup
- the observable belongs to the right-generator domain
Why these conditions cannot be dropped
- outside D(L) the defining difference quotient need not converge
- the result does not claim continuity, closedness, or equality with a differential expression
Proof route
- choose the generator limit carried by domain membership
- prove the chosen value is independent of the witness
- derive additivity and scalar compatibility from quotient limits
- bundle the result as a LinearMap
Lean interface notes
- the domain is generatorDomainSubmodule S
- the map is noncomputable because the limit witness is selected classically
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Lean statement
noncomputable def rightGenerator (S : ContinuousLinearSemigroup M) :
generatorDomainSubmodule S →ₗ[ℝ] M where
toFun := rightGeneratorValue S
map_add' := rightGeneratorValue_add S
map_smul' := rightGeneratorValue_smul S
/-- The canonical generator commutes with the semigroup on its invariant
domain. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:182published source at 7bcd37294df1