CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Strong continuity does not imply operator-norm continuity

Statement refuted

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $1 \leq p < \infty$ and let $(T (t))_{t \geq 0}$ be the right-translation semigroup on $L^{p} (R)$ (The right-translation semigroup on Lp has the weak derivative as generator), which is strongly continuous. Then $T$ is not continuous at $0$ in the operator norm: $∥ T (t) - I ∥ \geq 2^{1 / p}$ for every $t > 0$ , so $∥ T (t) - I ∥ \to̸ 0$ as $t ↓ 0$ . Hence strong continuity of a $C_{0}$ -semigroup is strictly weaker than norm continuity of $t \mapsto T (t)$ .

Refuted claim. For a strongly continuous semigroup on a Banach space, the map $t \mapsto T (t)$ is continuous at $0$ in the operator norm. The right-translation semigroup on $L^{p} (R)$ , $1 \leq p < \infty$ , is strongly continuous, but the distance $∥ T (t) - I ∥$ stays bounded below by $2^{1 / p}$ for all $t > 0$ .

Facts & Assumptions

Given: Countable Choice; $1 \leq p < \infty$ ; the right-translation semigroup $(T (t))_{t \geq 0}$ on $L^{p} (R)$ with $(T (t) g) (s) = g (s + t)$ , which is a strongly continuous semigroup of isometries (The right-translation semigroup on Lp has the weak derivative as generator, Strongly continuous semigroup); for $t > 0$ the function $f_{t} : = t^{- 1 / p} 1_{(0, t)}$ .

[F1]

$(T (t) g) (s) = g (s + t)$ for $g \in L^{p} (R)$ , so $T (t)$ acts by translation of the argument; translation preserves almost-everywhere classes (The right-translation semigroup on Lp has the weak derivative as generator, Translation of a function on $R^{n}$ ).

[F2]

For $1 \leq p < \infty$ the class norm is $∥ h ∥_{p} = (\int_{R} ∣ h (s) ∣^{p} d s)^{1 / p}$ , and indicators of sets of finite measure have the $p$ -th power of the norm equal to the measure of the set; null sets are invisible (The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

The operator norm is the supremum of $∥ A h ∥$ over the unit vectors $h$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces).

Counterexample

technique · direct computation with the unit vector $f_t=t^{-1/p}\mathbf 1_{(0,t)}$

1.1F2

For every $t > 0$ the vector $f_{t}$ has norm $∥ f_{t} ∥_{p}^{p} = \int_{0}^{t} t^{- 1} d s = 1$ , so $f_{t}$ is a unit vector of $L^{p} (R)$ .

2.1F1step 1.1

$T (t) f_{t} = t^{- 1 / p} 1_{(- t, 0)}$ : indeed $(T (t) f_{t}) (s) = f_{t} (s + t) = t^{- 1 / p} 1_{(0, t)} (s + t)$ , and $s + t \in (0, t)$ exactly when $s \in (- t, 0)$ .

3.1F2step 2.1

The two indicators $1_{(- t, 0)}$ and $1_{(0, t)}$ are disjoint up to the null set ${0}$ , so $∥ T (t) f_{t} - f_{t} ∥_{p}^{p} = t^{- 1} \int_{R} ∣ 1_{(- t, 0)} - 1_{(0, t)} ∣^{p} d s = t^{- 1} (λ ((- t, 0)) + λ ((0, t))) = t^{- 1} (t + t) = 2$ ; hence $∥ T (t) f_{t} - f_{t} ∥_{p} = 2^{1 / p}$ .

4.1F3step 1.1step 3.1∎

Since $f_{t}$ is a unit vector, [F3] and [step 3.1] give $∥ T (t) - I ∥ \geq ∥ T (t) f_{t} - f_{t} ∥_{p} = 2^{1 / p}$ for every $t > 0$ , so $∥ T (t) - I ∥$ does not tend to $0$ as $t ↓ 0$ and the semigroup is not continuous at $0$ in the operator norm, although it is strongly continuous; hence strong continuity does not imply operator-norm continuity.

Depends on

Used by

Nothing in the library uses this result yet.

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)