Semigroups with nonquasianalytic growth
Volume 104 / 1993
Abstract
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.
