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Yosida approximants are bounded and converge on the domain

Statement

Under the hypotheses of The Yosida resolvent converges strongly to the identity, let $A_{λ} = λ A R (λ, A)$ be the Yosida approximants (Yosida approximants). Then $∥ A_{λ} ∥ \leq λ^{2} \frac{M}{λ - ω} + ∣ λ ∣$ for $λ > ω$ , and $A_{λ} x \to A x$ as $λ \to \infty$ for every $x \in D (A)$ .

Facts & Assumptions

Given: The hypotheses of The Yosida resolvent converges strongly to the identity on the closed densely defined operator $A$ , and the Yosida approximants $A_{λ} = λ A R (λ, A) = λ^{2} R (λ, A) - λ I$ (Yosida approximants).

[F1]

$A_{λ} = λ^{2} R (λ, A) - λ I$ is bounded with $∥ A_{λ} ∥ \leq λ^{2} ∥ R (λ, A) ∥ + ∣ λ ∣$ (Yosida approximants, The operator norm is a norm on the space of bounded linear operators, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F2]

$∥ R (λ, A) ∥ \leq M / (λ - ω)$ and $λ R (λ, A) x \to x$ for all $x \in X$ , $λ R (λ, A) A x \to A x$ for $x \in D (A)$ (the bound is a hypothesis of The Yosida resolvent converges strongly to the identity, which proves both convergence conclusions).

[F3]

For $x \in D (A)$ one has $A R (λ, A) x = R (λ, A) A x$ : both equal $λ R (λ, A) x - x$ by the resolvent identity (Resolvent and spectrum of a closed operator on a Banach space).

Proof

technique · direct: the norm bound from the identity form, and pointwise convergence from the strong resolvent convergence

1.1F1F2

By [F1] and [F2], $∥ A_{λ} ∥ \leq λ^{2} \frac{M}{λ - ω} + ∣ λ ∣$ for $λ > ω$ , which is the asserted bound.

1.2F2F3

For $x \in D (A)$ , $A_{λ} x = λ A R (λ, A) x = λ R (λ, A) A x$ by [F3]; by [F2] the right-hand side converges to $A x$ as $λ \to \infty$ . Hence $A_{λ} x \to A x$ for every $x \in D (A)$ .

2.1step 1.1step 1.2∎

Both conclusions hold for every $λ > ω$ , and no uniform convergence on all of $D (A)$ or on bounded sets is claimed.

Depends on

Used by

Bounded Yosida semigroups converge to the generated semigroup Theorem

Dependency tree · two levels

23 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)