Zero-two law for cosine families

Abstract

For ${\left(C(t)\right)}_{t\ge 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate ${\mathrm{limsup}}_{t\to 0^{+}}\Vert C(t)-I\Vert <2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of ${\sup }_{t\ge 0}\Vert C(t)-I\Vert <2$ yields that $C(t)=I$ for all $t\ge 0$. Additionally, we derive alternative proofs for similar results for $C_0$-semigroups.

10 pages, changes to previous version: Major changes and rearrangements in the set-up to improve readability. Theorem 1.1 now only deals with the '$\mathrm{limsup}$' assertion, whereas the '$\sup$' assertion is moved to Section 3. The 'scaled versions' of the result were discarded, see also the new remark after eq. (1.8). for a correction of some intially stated claim

References in corpus (1)

• Regularity of Semigroups via the Asymptotic Behaviour at Zero

Cited by in corpus (1)

• On cosine families close to scalar cosine families