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Lumer-Phillips 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 densely defined dissipative operator on a Banach space $X$ (Dissipative operator). Then the following are equivalent: (a) $A$ generates a strongly continuous semigroup of contractions; (b) $R a n (λ_{0} I - A) = X$ for some $λ_{0} > 0$ ; (c) $R a n (λ I - A) = X$ for every $λ > 0$ . In that case $A$ is closed, $(0, \infty) \subseteq ρ (A)$ , $∥ R (λ, A) ∥ \leq 1 / λ$ for all $λ > 0$ , and $A$ is maximal dissipative (it has no proper dissipative extension).

Facts & Assumptions

Given: Dependent Choice; A densely defined dissipative operator $A : D (A) \subseteq X \to X$ on a Banach space $X$ (Dissipative operator, Strongly continuous semigroup).

[F1]

Dissipativity means $∥ (λ I - A) x ∥ \geq λ ∥ x ∥$ for all $λ > 0$ and $x \in D (A)$ ; hence each $λ I - A$ is injective and $∥ (λ I - A)^{- 1} z ∥ \leq λ^{- 1} ∥ z ∥$ on the range of $λ I - A$ (Dissipative operator).

[F2]

Contraction Hille-Yosida: a closed densely defined operator with $(0, \infty) \subseteq ρ (A)$ and $∥ R (λ, A) ∥ \leq 1 / λ$ for all $λ > 0$ generates a strongly continuous semigroup of contractions; conversely the generator of a contraction semigroup has $(0, \infty) \subseteq ρ (A)$ and $∥ R (λ, A) ∥ \leq 1 / λ$ (Contraction Hille-Yosida theorem).

[F3]

Resolvent identity: $R (λ, A) - R (λ_{0}, A) = (λ_{0} - λ) R (λ, A) R (λ_{0}, A)$ (Resolvent identity for closed operators). Consequently, for $∣ λ_{0} - λ ∣ < 1 / ∥ R (λ_{0}, A) ∥$ the series $\sum_{k \geq 0} (λ_{0} - λ)^{k} R (λ_{0}, A)^{k + 1}$ converges in $B (X)$ and its sum is the inverse of $λ I - A$ : writing $R_{0} = R (λ_{0}, A)$ and $Q = I + (λ - λ_{0}) R_{0}$ , one has $(λ I - A) R_{0} = Q$ and, on $D (A)$ , $λ I - A = Q (λ_{0} I - A)$ . Thus $R_{0} Q^{- 1}$ is a two-sided inverse and has range in $D (A)$ . In the Neumann series $Q^{- 1} = \sum_{k \geq 0} (λ_{0} - λ)^{k} R_{0}^{k}$ , the operators commute with $R_{0}$ by taking limits of polynomials, giving the displayed series. Indeed, for $X \neq {0}$ , $B (X)$ is complete for the operator norm (If (Y) is Banach then (\mathcal B(X,Y)) is Banach) with submultiplicative composition (Composition satisfies |ST|\le|S|,|T|, The operator norm as the least bound and as the unit-sphere or unit-ball supremum), so the Neumann-series computation applies — for complex $X$ through Neumann series and the unital Banach-algebra structure (Unital Banach algebra), and for real $X$ by the identical telescoping computation. An operator with a bounded everywhere-defined inverse is closed: the inverse graph is the zero set of the continuous map $(z, y) \mapsto y - R z$ , and swapping graph coordinates gives the graph of the original operator. A scalar shift of its graph is a homeomorphism. Hence $A$ is closed as soon as $λ I - A$ has such an inverse; a closed bijective operator with bounded inverse lies in the resolvent set (Resolvent and spectrum of a closed operator on a Banach space).

[F4]

Operators of the form $λ I - A$ with $λ > 0$ are closed when $A$ is closed, and $A$ is closed as soon as some $λ I - A$ has a bounded everywhere-defined inverse; generators are closed and densely defined (The generator is closed and densely defined, Resolvent and spectrum of a closed operator on a Banach space).

Proof

technique · direct: dissipativity gives injectivity and the inverse bound, surjectivity at one $\lambda_0$ propagates by the resolvent series, and the contraction generation theorem finishes

1.1F1F4

If $X = {0}$ then $D (A) = X$ and all claims hold for the unique zero operator and semigroup; hence assume $X \neq {0}$ . (b) $\Rightarrow$ closedness, $λ_{0} \in ρ (A)$ . Assume $Ran (λ_{0} I - A) = X$ for some $λ_{0} > 0$ . By [F1] $λ_{0} I - A$ is injective with $∥ (λ_{0} I - A)^{- 1} z ∥ \leq λ_{0}^{- 1} ∥ z ∥$ for all $z \in X$ ; thus its inverse is a bounded everywhere-defined operator and $λ_{0} I - A$ is bijective, so $λ_{0} \in ρ (A)$ and $A$ is closed by [F4].

1.2F2

(a) $\Rightarrow$ (b),(c). If $A$ generates a contraction semigroup, [F2] gives $(0, \infty) \subseteq ρ (A)$ with $∥ R (λ, A) ∥ \leq 1 / λ$ ; therefore every $λ I - A$ , $λ > 0$ , is bijective onto $X$ , which is (c) and, taking e.g. $λ = 1$ , also (b). Trivially (c) $\Rightarrow$ (b).

2.1F1F3step 1.1

Propagation to $(0, \infty)$ . With $λ_{0} \in ρ (A)$ and $∥ R (λ_{0}, A) ∥ \leq 1 / λ_{0}$ , the series of [F3] converges for $∣ λ - λ_{0} ∣ < λ_{0}$ and represents $R (λ, A)$ ; hence $(0, 2 λ_{0}) \subseteq ρ (A)$ . Dissipativity now gives $∥ R (λ, A) ∥ \leq 1 / λ$ for every $λ \in ρ (A) \cap (0, \infty)$ by [F1], in particular on $(0, 2 λ_{0})$ . Replacing $λ_{0}$ by any $λ_{1} \in (λ_{0}, 2 λ_{0})$ , the same argument gives $(0, 2 λ_{1}) \subseteq ρ (A)$ with $λ_{1} > λ_{0}$ ; iterating with the explicit points $λ_{k} = (3 / 2)^{k} λ_{0}$ , each inside the previous interval $(0, 2 λ_{k - 1})$ , gives $(0, 2 λ_{k}) \subseteq ρ (A)$ for all $k$ . Since $λ_{k} \to \infty$ , this yields $(0, \infty) \subseteq ρ (A)$ and $∥ R (λ, A) ∥ \leq 1 / λ$ for all $λ > 0$ . In particular $λ I - A$ is surjective for every $λ > 0$ , so (c) holds.

3.1F2step 1.1step 2.1

(b) $\Rightarrow$ (a). $A$ is closed and densely defined by hypothesis and [step 1.1]; [step 2.1] supplies $(0, \infty) \subseteq ρ (A)$ and $∥ R (λ, A) ∥ \leq 1 / λ$ ; hence [F2] makes $A$ the generator of a strongly continuous semigroup of contractions.

3.2F1step 2.1

Maximal dissipativity. Let $A^{'} \supseteq A$ be a dissipative extension and fix $λ > 0$ . By [step 2.1], $λ I - A$ maps $D (A)$ onto $X$ ; given $x \in D (A^{'})$ , choose $y \in D (A)$ with $(λ I - A) y = (λ I - A^{'}) x$ . Since $A^{'}$ agrees with $A$ on $D (A)$ , $(λ I - A^{'}) y = (λ I - A) y = (λ I - A^{'}) x$ , and injectivity of $λ I - A^{'}$ by [F1] gives $x = y \in D (A)$ . Hence $D (A^{'}) = D (A)$ and $A^{'} = A$ : $A$ has no proper dissipative extension.

4.1step 1.1step 2.1step 3.1step 1.2step 3.2∎

Combining the implications: (a), (b) and (c) are equivalent for a densely defined dissipative operator, and in that case $A$ is closed, $(0, \infty) \subseteq ρ (A)$ , $∥ R (λ, A) ∥ \leq 1 / λ$ , and $A$ is maximal dissipative.

Depends on

Used by

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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)
Mathew A. Johnson, Math 951 Lecture Notes, Chapter 6: Introduction to Semigroup Methods, University of Kansas (complete 37-page chapter) (standard reference, not scraped)