The bounded perturbation theorem

Adding a bounded operator B to the generator A of a C₀-semigroup leaves the generator property intact: the domain does not move, so the perturbed operator is B +ᵥ A, and the growth exponent increases by at most M ‖B‖ when the original growth constant is M.

• On the operator side, TauCeti.Semigroups.IsMDissipative.subScalar_vadd says that the Lumer--Phillips hypotheses survive the perturbation once the spectral parameter is shifted by ‖B‖. Undoing the shift with TauCeti.Semigroups.StronglyContinuousSemigroup.expShift turns the resulting contraction semigroup into one of growth (‖B‖, 1) generated by B +ᵥ A.
• On the semigroup side, the equivalent norm |||x||| = sup (t ≥ 0), exp (-omega t) ‖S(t)x‖ makes the exponentially shifted semigroup a contraction. Applying the operator theorem there and transporting back gives the general growth bound (omega + M ‖B‖, M).

Main results

• TauCeti.Semigroups.IsMDissipative.exists_stronglyContinuousSemigroup_generator_eq_vadd: a bounded perturbation of a densely defined m-dissipative operator generates a C₀-semigroup of growth (‖B‖, 1).
• TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd: the bounded perturbation theorem at a general growth bound (omega, M).
• TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd_of_hasGrowthBound_one: the quasi-contractive specialization.
• TauCeti.Semigroups.ContractionSemigroup.exists_generator_eq_vadd: the contraction case.

References

Engel--Nagel, One-Parameter Semigroups for Linear Evolution Equations, Theorem III.1.3; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 3, Theorem 1.1.

The equivalent growth norm

theorem TauCeti.Semigroups.IsMDissipative.exists_stronglyContinuousSemigroup_generator_eq_vadd {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {A : X →ₗ.[ℝ] X} [CompleteSpace X] (hA : IsMDissipative A) (hdense : Dense ↑A.domain) (B : X →L[ℝ] X) : ∃ (S : StronglyContinuousSemigroup X), S.generator = ↑B +ᵥ A ∧ S.HasGrowthBound ‖B‖ 1

The bounded perturbation theorem, operator form. A bounded perturbation B +ᵥ A of a densely defined m-dissipative operator A generates a strongly continuous semigroup with growth bound (‖B‖, 1).

theorem TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (S : StronglyContinuousSemigroup X) {omega M : ℝ} (hb : S.HasGrowthBound omega M) (B : X →L[ℝ] X) : ∃ (T : StronglyContinuousSemigroup X), T.generator = ↑B +ᵥ S.generator ∧ T.HasGrowthBound (omega + M * ‖B‖) M

The bounded perturbation theorem. If a strongly continuous semigroup S has growth bound (omega, M) and B is bounded, then B +ᵥ S.generator generates a strongly continuous semigroup with growth bound (omega + M * ‖B‖, M).

theorem TauCeti.Semigroups.StronglyContinuousSemigroup.exists_generator_eq_vadd_of_hasGrowthBound_one {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (S : StronglyContinuousSemigroup X) {omega : ℝ} (hb : S.HasGrowthBound omega 1) (B : X →L[ℝ] X) : ∃ (T : StronglyContinuousSemigroup X), T.generator = ↑B +ᵥ S.generator ∧ T.HasGrowthBound (omega + ‖B‖) 1

The bounded perturbation theorem. If a strongly continuous semigroup S has growth bound (omega, 1) and B is bounded, then B +ᵥ S.generator generates a strongly continuous semigroup with growth bound (omega + ‖B‖, 1).

theorem TauCeti.Semigroups.ContractionSemigroup.exists_generator_eq_vadd {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (S : ContractionSemigroup X) (B : X →L[ℝ] X) : ∃ (T : StronglyContinuousSemigroup X), T.generator = ↑B +ᵥ S.generator ∧ T.HasGrowthBound ‖B‖ 1

The bounded perturbation theorem for contraction semigroups. A bounded perturbation of the generator of a contraction semigroup generates a strongly continuous semigroup with growth bound (‖B‖, 1).