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The Yosida resolvent converges strongly to the identity

Statement

Let $A : D (A) \subseteq X \to X$ be closed and densely defined on a Banach space $X$ , and let $M \geq 1$ , $ω \in R$ be such that $(ω, \infty) \subseteq ρ (A)$ and $∥ R (λ, A)^{n} ∥ \leq M (λ - ω)^{- n}$ for all real $λ > ω$ and every $n \geq 1$ (Resolvent and spectrum of a closed operator on a Banach space). Then, as $λ \to \infty$ along the reals, $λ R (λ, A) x \to x$ for every $x \in X$ , and $λ R (λ, A) A x \to A x$ for every $x \in D (A)$ .

Facts & Assumptions

Given: A closed densely defined operator $A$ on a Banach space $X$ with $(ω, \infty) \subseteq ρ (A)$ and $∥ R (λ, A)^{n} ∥ \leq M (λ - ω)^{- n}$ for all real $λ > ω$ and $n \geq 1$ (Resolvent and spectrum of a closed operator on a Banach space).

[F1]

Resolvent identities: $R (λ, A) X = D (A)$ , $R (λ, A) (λ I - A) y = y$ for $y \in D (A)$ , and $λ R (λ, A) x - x = A R (λ, A) x$ for $x \in X$ ; in particular $λ R (λ, A) y = y + R (λ, A) A y$ for $y \in D (A)$ , since $λ R (λ, A) y = R (λ, A) ((λ I - A) y + A y) = y + R (λ, A) A y$ (Resolvent and spectrum of a closed operator on a Banach space).

[F2]

First power estimate: $∥ R (λ, A) ∥ \leq M / (λ - ω)$ for $λ > ω$ (Resolvent power estimates for semigroup generators is the source of this estimate in the semigroup case; here it is assumed directly).

[F3]

$D (A)$ is dense in $X$ , and operators of the form $λ R (λ, A)$ are bounded (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Proof

technique · direct: the identity on the dense domain and a uniform bound, then a three-epsilon argument

1.1F1F2

For $y \in D (A)$ and $λ > ω$ , [F1] gives $λ R (λ, A) y = y + R (λ, A) A y$ , hence by [F2] $∥ λ R (λ, A) y - y ∥ \leq \frac{M}{λ - ω} ∥ A y ∥ \to 0$ as $λ \to \infty$ .

1.2F2

The operators $B_{λ} : = λ R (λ, A)$ are uniformly norm bounded for large $λ$ : $∥ B_{λ} ∥ \leq \frac{λ M}{λ - ω} \leq 2 M$ for $λ \geq 2 ω$ when $ω > 0$ , and $∥ B_{λ} ∥ \leq M$ for $ω \leq 0$ and $λ > 0$ .

2.1F3step 1.1step 1.2

For arbitrary $x \in X$ : given $ε > 0$ , choose $y \in D (A)$ with $∥ x - y ∥ < ε / (4 M + 2)$ by density [F3]; by [step 1.1] choose $λ_{0}$ with $∥ λ R (λ, A) y - y ∥ < ε / 2$ for $λ > λ_{0}$ ; then for such $λ$ , $∥ λ R (λ, A) x - x ∥ \leq ∥ B_{λ} (x - y) ∥ + ∥ λ R (λ, A) y - y ∥ + ∥ y - x ∥ < 2 M \frac{ε}{4 M + 2} + \frac{ε}{2} + \frac{ε}{4 M + 2} < ε$ . Hence $λ R (λ, A) x \to x$ for every $x \in X$ .

3.1F1step 2.1∎

The second statement is [step 2.1] applied to the vector $A x \in X$ : $λ R (λ, A) A x \to A x$ ; by [F1] and [F2] this is the same as $λ A R (λ, A) x = λ^{2} R (λ, A) x - λ x \to A x$ for $x \in D (A)$ .

Depends on

Used by

Yosida approximants are bounded and converge on the domain Lemma

Dependency tree · two levels

29 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)