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## Studia Mathematica

1993 | 104 | 3 | 229-241
Tytuł artykułu

### Semigroups with nonquasianalytic growth

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
229-241
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-02-03
Twórcy
autor
• Institute of Mathematics, P.O. Box 631, 10000 Hanoi, Vietnam.
Bibliografia
• [1] G. R. Allan and T. J. Ransford, Power-dominated elements in a Banach algebra, Studia Math. 94 (1989), 63-79.
• [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.
• [3] I. C. Gokhberg and M. G. Kreǐn, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspekhi Mat. Nauk 12 (2) (1957), 43-118; English transl.: Amer. Math. Soc. Transl. (2) 13 (1960), 185-264.
• [4] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc., Providence, R.I., 1957.
• [5] Y. Katznelson, An Introduction to Harmonic Analysis, 2nd ed., Dover, New York 1976.
• [6] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
• [7] Yu. I. Lyubich, V. I. Matsaev and G. M. Fel'dman, Representations with separable spectrum, Funct. Anal. Appl. 7 (1973), 129-136.
• [8] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42.
• [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin 1983.
• [10] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Univ. Press, Oxford 1968.
• [11] Vũ Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84.
• [12] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semidynamical systems, J. Differential Equations, to appear.
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Bibliografia
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