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1993 | 104 | 3 | 229-241
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Semigroups with nonquasianalytic growth

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Abstrakty
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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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
104
Numer
3
Strony
229-241
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-02-03
Twórcy
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv104i3p229bwm
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