The duality-map characterization of dissipativity

On a real normed space X, an unbounded operator A : X →ₗ.[ℝ] X is dissipative in the resolvent-range sense of TauCeti.Semigroups.IsDissipative,

lambda * ‖x‖ ≤ ‖lambda • x - A x‖ for all lambda > 0 and x ∈ D(A),

exactly when for every x ∈ D(A) some element x' of the duality set J(x) = {x' | x' x = ‖x‖², ‖x'‖ = ‖x‖} satisfies x' (A x) ≤ 0 (isDissipative_iff_exists_mem_dualitySet_apply_nonpos). This is the Banach-space counterpart of the Hilbert condition ⟪A x, x⟫ ≤ 0 and the form in which dissipativity is usually verified for concrete operators on Lᵖ, C₀ or ℓᵖ, where the duality set is explicit.

For the generator of a contraction semigroup the inequality holds for every element of the duality set (ContractionSemigroup.apply_generator_nonpos_of_mem_dualitySet).

References

• K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Proposition II.3.23.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 1, Theorem 4.2 and Theorem 4.3.

theorem TauCeti.Semigroups.IsDissipative.exists_mem_dualitySet_apply_nonpos {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {A : X →ₗ.[ℝ] X} (hA : IsDissipative A) (x : ↥A.domain) : ∃ f ∈ dualitySet ℝ ↑x, f (↑A x) ≤ 0

Dissipative operators are dissipative in the duality-map sense. If A is dissipative, then every x ∈ D(A) has an element x' of its duality set with x' (A x) ≤ 0.

theorem TauCeti.Semigroups.isDissipative_iff_exists_mem_dualitySet_apply_nonpos {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] (A : X →ₗ.[ℝ] X) : IsDissipative A ↔ ∀ (x : ↥A.domain), ∃ f ∈ dualitySet ℝ ↑x, f (↑A x) ≤ 0

The duality-map characterization of dissipativity. An unbounded operator on a real normed space is dissipative exactly when every x ∈ D(A) has an element x' of its duality set J(x) with x' (A x) ≤ 0.

theorem TauCeti.Semigroups.ContractionSemigroup.apply_generator_nonpos_of_mem_dualitySet {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] (S : ContractionSemigroup X) (x : ↥S.generator.domain) {f : StrongDual ℝ X} (hf : f ∈ dualitySet ℝ ↑x) : f (↑S.generator x) ≤ 0

The generator of a contraction semigroup is dissipative for every element of the duality set: x' (A x) ≤ 0 whenever x ∈ D(A) and x' ∈ J(x). This strengthens the existential condition that characterizes general dissipative operators (isDissipative_iff_exists_mem_dualitySet_apply_nonpos).