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Hille-Yosida generation theorem

Statement

Assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $A : D (A) \subseteq X \to X$ be a closed and densely defined linear operator on a Banach space $X$ and let $M \geq 1$ , $ω \in R$ . Then $A$ generates a strongly continuous semigroup $(T (t))_{t \geq 0}$ with $∥ T (t) ∥ \leq M e^{ω t}$ for all $t \geq 0$ if and only if both: (i) $(ω, \infty) \subseteq ρ (A)$ , and (ii) $∥ R (λ, A)^{n} ∥ \leq M (λ - ω)^{- n}$ for every real $λ > ω$ and every $n \geq 1$ (Resolvent and spectrum of a closed operator on a Banach space). All resolvent powers are required in general; the first power alone guarantees all power estimates when $M = 1$ , the exponentially rescaled contraction case, where all higher power estimates follow from the single one by submultiplicativity (treated later on this page).

Facts & Assumptions

Given: Dependent Choice; A closed densely defined linear operator $A$ on a Banach space $X$ and constants $M \geq 1$ , $ω \in R$ (Resolvent and spectrum of a closed operator on a Banach space, The generator is closed and densely defined).

[F1]

Sufficiency: under (i) $(ω, \infty) \subseteq ρ (A)$ and (ii) $∥ R (λ, A)^{n} ∥ \leq M (λ - ω)^{- n}$ for all real $λ > ω$ , $n \geq 1$ , the operator $A$ generates a strongly continuous semigroup with $∥ T (t) ∥ \leq M e^{ω t}$ (Bounded Yosida semigroups converge to the generated semigroup).

[F2]

Necessity of the location of the spectrum and of the first estimate: if $T$ is a strongly continuous semigroup with generator $A$ and $∥ T (t) ∥ \leq M e^{ω t}$ , then $A$ is closed and densely defined, $(ω, \infty) \subseteq ρ (A)$ , and $∥ R (λ, A) ∥ \leq M / (λ - ω)$ (The generator is closed and densely defined, Laplace transform formula for the resolvent, Strongly continuous semigroup).

[F3]

Necessity of all powers: under the hypotheses of [F2], $R (λ, A)^{m} x = \frac{1}{(m - 1)!} \int_{0}^{\infty} s^{m - 1} e^{- λ s} T (s) x d s$ and $∥ R (λ, A)^{m} ∥ \leq M (λ - ω)^{- m}$ for every $m \geq 1$ (Resolvent power estimates for semigroup generators).

Proof

technique · direct: sufficiency is the Yosida construction and necessity is read off from the Laplace representation of the resolvent

1.1F1

(Sufficiency.) Assume (i) and (ii). Then all hypotheses of [F1] hold, so $A$ generates a strongly continuous semigroup $(T (t))_{t \geq 0}$ with $∥ T (t) ∥ \leq M e^{ω t}$ for all $t \geq 0$ .

1.2F2

(Necessity, domain and spectrum.) Assume conversely that $A$ generates $T$ with $∥ T (t) ∥ \leq M e^{ω t}$ . Then $A$ is closed with dense domain by [F2]; the Laplace-transform formula for the resolvent gives $λ \in ρ (A)$ and $∥ R (λ, A) ∥ \leq M / (λ - ω)$ for every real $λ > ω$ ; in particular $(ω, \infty) \subseteq ρ (A)$ , which is (i).

1.3F3

(Necessity, all powers.) Under the same hypothesis, [F3] gives the integral representation of every power $R (λ, A)^{m}$ and the estimate $∥ R (λ, A)^{m} ∥ \leq M (λ - ω)^{- m}$ for all $m \geq 1$ and real $λ > ω$ , which is (ii).

2.1step 1.1step 1.2step 1.3∎

Combining [step 1.1] with [steps 1.2-1.3]: $A$ generates a strongly continuous semigroup with $∥ T (t) ∥ \leq M e^{ω t}$ if and only if (i) and (ii) hold. The general theorem retains all power estimates; the case where the first estimate alone suffices ( $M = 1$ , $ω = 0$ ) is isolated as the next corollary.

Depends on

Used by

Contraction Hille-Yosida theorem Corollary
A first resolvent estimate does not ensure the prescribed semigroup bound Counterexample
Semigroup sign and generator conventions Remark

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Klaus-Jochen Engel and Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194 (complete author-hosted monograph) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations, Karlsruhe Institute of Technology (2023/24 course, complete lecture notes) (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer 2011 (complete 614-page text) (standard reference, not scraped)