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.
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We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.
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