The Hille--Yosida theorem for contraction and quasi-contraction semigroups

When the constant in the growth bound is M = 1, the Hille--Yosida resolvent-power estimates ‖R(lambda, A) ^ n‖ ≤ 1 / (lambda - omega) ^ n for all n ≥ 1 are implied by the single estimate ‖R(lambda, A)‖ ≤ 1 / (lambda - omega), by submultiplicativity of the operator norm. This gives the classical form of the generation theorem, in which only the first power of the resolvent is controlled:

• a densely defined operator A on a real Banach space generates a C₀-semigroup with ‖S(t)‖ ≤ e^{omega t} exactly when (omega, ∞) lies in its resolvent set and ‖R(lambda, A)‖ ≤ 1 / (lambda - omega) for lambda > omega;
• in particular A generates a contraction semigroup exactly when (0, ∞) lies in its resolvent set and ‖R(lambda, A)‖ ≤ 1 / lambda for lambda > 0.

The contraction form is obtained through the Lumer--Phillips characterization, using that these resolvent conditions are equivalent to m-dissipativity.

Main results

• TauCeti.Semigroups.exists_generator_eq_and_hasGrowthBound_one_iff_forall_norm_resolvent_le: the Hille--Yosida characterization of quasi-contraction semigroups.
• TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff_forall_norm_resolvent_le: the Hille--Yosida characterization of contraction semigroups.

References

• K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Theorem II.3.5 and Corollary II.3.6.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 1, Theorem 3.1 and Corollary 3.8.

theorem TauCeti.Semigroups.exists_generator_eq_and_hasGrowthBound_one_iff_forall_norm_resolvent_le {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (A : X →ₗ.[ℝ] X) (omega : ℝ) : (∃ (S : StronglyContinuousSemigroup X), S.generator = A ∧ S.HasGrowthBound omega 1) ↔ Dense ↑A.domain ∧ (∀ (lambda : ℝ), omega < lambda → lambda ∈ A.resolventSet) ∧ ∀ (lambda : ℝ), omega < lambda → ‖A.resolvent lambda‖ ≤ (lambda - omega)⁻¹

Hille--Yosida for quasi-contraction semigroups. An unbounded operator A on a real Banach space generates a strongly continuous semigroup with ‖S(t)‖ ≤ e^{omega t} if and only if its domain is dense, the half-line (omega, ∞) lies in its resolvent set, and ‖R(lambda, A)‖ ≤ 1 / (lambda - omega) for every lambda > omega.

theorem TauCeti.Semigroups.exists_contractionSemigroup_generator_eq_iff_forall_norm_resolvent_le {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (A : X →ₗ.[ℝ] X) : (∃ (S : ContractionSemigroup X), S.generator = A) ↔ Dense ↑A.domain ∧ (∀ (lambda : ℝ), 0 < lambda → lambda ∈ A.resolventSet) ∧ ∀ (lambda : ℝ), 0 < lambda → ‖A.resolvent lambda‖ ≤ lambda⁻¹

Hille--Yosida for contraction semigroups. An unbounded operator A on a real Banach space generates a strongly continuous contraction semigroup if and only if its domain is dense, every lambda > 0 lies in its resolvent set, and ‖R(lambda, A)‖ ≤ 1 / lambda for every lambda > 0.