Semigroups with nonquasianalytic growth

We study asymptotic behavior of ${\displaystyle C_0}$-semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then ${\displaystyle \underset{t\to \infty }{lim}\frac{1}{\alpha }\left(t\right)\parallel T\left(t\right)x\parallel =0}$, ∀x ∈ X. If, moreover, f is a function in ${\displaystyle L^1\_\{\alpha \}\left({ℝ}_{+}\right)}$ which is of spectral synthesis in a corresponding algebra ${\displaystyle L^1\_\left\{{\alpha }_1\right\}(ℝ)}$ with respect to (iσ(A)) ∩ ℝ, then ${\displaystyle \underset{t\to \infty }{lim}\frac{1}{\alpha }\left(t\right)\parallel T\left(t\right)f̂\left(T\right)\parallel =0}$, where ${\displaystyle f̂\left(T\right)={ʃ}_0^{\infty }f\left(t\right)T\left(t\right)dt}$. Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of Katznelson-Tzafriri, Lyubich-Vũ Quôc Phóng, Arendt-Batty, ..., concerning contraction semigroups. The proofs are based on the operator form of the Tauberian Theorem for Beurling algebras with nonquasianalytic weight.