Geodesic
Semigroups with nonquasianalytic growth
Studia Mathematica, Tome 104 (1993) no. 3, pp. 229-241

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We study asymptotic behavior of $C_0$-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 $lim_{t→∞} 1/α(t)∥T(t)x∥ = 0$, ∀x ∈ X. If, moreover, f is a function in $L^{1}_{α}(ℝ_{+})$ which is of spectral synthesis in a corresponding algebra $L^{1}_{α_1}(ℝ)$ with respect to (iσ(A)) ∩ ℝ, then $lim_{t→∞} 1/α(t) ∥T(t)f̂(T)∥ = 0$, where $f̂(T) = ʃ_{0}^{∞} f(t)T(t)dt$. 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. 
Vũ Quôc Phóng. Semigroups with nonquasianalytic growth. Studia Mathematica, Tome 104 (1993) no. 3, pp. 229-241. doi: 10.4064/sm-104-3-229-241
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