The Hilbert-space characterization of dissipativity

On a real inner-product space the Banach-space definition of dissipativity, lambda * ‖x‖ ≤ ‖lambda • x - A x‖ for all lambda > 0, collapses to the familiar inner-product condition

⟪A x, x⟫ ≤ 0 for all x ∈ D(A).

Both implications come from the polarization identity ‖lambda • x - A x‖² = (lambda ‖x‖)² - 2 lambda ⟪x, A x⟫ + ‖A x‖²: the sign condition makes the correction term nonnegative, and conversely dividing the resulting inequality 2 lambda ⟪A x, x⟫ ≤ ‖A x‖² by lambda and letting lambda → ∞ forces ⟪A x, x⟫ ≤ 0.

Combined with TauCeti.Semigroups.ContractionSemigroup.isDissipative_generator, this gives the Hilbert-space form of the converse of the Lumer--Phillips theorem: the generator of a contraction semigroup on a real Hilbert space satisfies ⟪A x, x⟫ ≤ 0.

References

Engel--Nagel, One-Parameter Semigroups for Linear Evolution Equations, Example II.3.24 and Proposition II.3.23; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 1, Section 4.

theorem TauCeti.Semigroups.isDissipative_iff_real_inner_nonpos {X : Type u_1} [NormedAddCommGroup X] [InnerProductSpace ℝ X] (A : X →ₗ.[ℝ] X) : IsDissipative A ↔ ∀ (x : ↥A.domain), inner ℝ (↑A x) ↑x ≤ 0

Dissipativity on a real inner-product space. An unbounded operator is dissipative exactly when ⟪A x, x⟫ ≤ 0 on its domain.

theorem TauCeti.Semigroups.ContractionSemigroup.real_inner_generator_nonpos {X : Type u_1} [NormedAddCommGroup X] [InnerProductSpace ℝ X] [CompleteSpace X] (S : ContractionSemigroup X) (x : ↥S.generator.domain) : inner ℝ (↑S.generator x) ↑x ≤ 0

The converse of the Lumer--Phillips theorem on a Hilbert space. The generator of a contraction semigroup on a real Hilbert space satisfies ⟪A x, x⟫ ≤ 0 on its domain.