Optimal decay of semi-uniformly stable operator semigroups with empty spectrum

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of the spectrum of its infinitesimal generator. More precisely, given an arbitrary positive function r r vanishing at ∞ ∞ , we construct a Banach space X X and a bounded semigroup ( T ( t ) ) t ≥ 0 (T(t))t ≥ 0 of operators on it whose infinitesimal generator A A has empty spectrum σ ( A ) = ∅ σ (A)= , but for which, for some x ∈ X x ∈ X , lim sup t → ∞ ‖ T ( t ) A − 1 x ‖ X r ( t ) = ∞ . {equation*} t→ ∞ {\|T(t)A⁻¹x\|X}{r(t)}=∞ . {equation*}