Resolvent norm decay does not characterize norm continuity

Abstract

We show that there exists a reflexive Banach space \( (\mathcal{X},\parallel .\parallel ) \) and a strongly continuous semigroup \( (\mathcal{T}(t))_{t \geqslant 0} \) with generator \( (\mathcal{A},\mathcal{D}(\mathcal{A})) \) on \( (\mathcal{X},\parallel .\parallel ) \) such that \( \lim _{\mu \in \mathbb{R},|\mu | \to \infty } \parallel R(i\mu ,\mathcal{A})\parallel = 0 \) but \( (\mathcal{T}(t))_{t \geqslant 0} \) is not eventually norm continuous. This answers a question of Amnon Pazy in the negative.