Abstract
For $\left(C(t)\right)_{t \geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\limsup_{t\to 0^{+}}\|C(t) - I\| <2$ 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 $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. Additionally, we derive alternative proofs for similar results for $C_{0}$-semigroups.
Highlights
Let (T (t))t≥0 denote a strongly continuous semigroup on the Banach space X with infinitesimal generator A
This has become known as zero-one law for semigroups
We study the zero-two law for strongly continuous cosine families on a Banach space, i.e. whether lim sup C(t) − I < 2 implies that lim sup C(t) − I = 0
Summary
Let (T (t))t≥0 denote a strongly continuous semigroup on the Banach space X with infinitesimal generator A. We study the zero-two law for strongly continuous cosine families on a Banach space, i.e. whether lim sup C(t) − I < 2 implies that lim sup C(t) − I = 0. The 0 − 3/2 law, i.e. holds for cosine families on general Banach spaces as was proved by Arendt in [1, Theorem 1.1 in Three Line Proofs]. Implies that T (t) = I for all t ≥ 0, see, e.g. Wallen [13] and Hirschfeld [8] This seems not to be well known among researchers working in the area of strongly continuous semigroup. We prove (1.6) for strongly continuous cosine families on Banach spaces This result is strongly motivated by the recent work of Bobrowski and Chojnacki. For λ ∈ ρ( A), R(λ, A) denotes (λI − A)−1
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