Semigroups with nonquasianalytic growth

Phóng Vũ

Studia Mathematica (1993)

  • Volume: 104, Issue: 3, page 229-241
  • ISSN: 0039-3223

Abstract

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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 l i m t → ∞ 1 / α ( t ) ∥ T ( t ) x ∥ = 0 , ∀x ∈ X. If, moreover, f is a function in L α 1 ( ℝ + ) which is of spectral synthesis in a corresponding algebra L α 1 1 ( ℝ ) with respect to (iσ(A)) ∩ ℝ, then l i m t → ∞ 1 / α ( t ) ∥ T ( t ) f ̂ ( T ) ∥ = 0 , where f ̂ ( T ) = ʃ 0 ∞ f ( t ) T ( t ) d t . 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.

How to cite

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Vũ, Phóng. "Semigroups with nonquasianalytic growth." Studia Mathematica 104.3 (1993): 229-241. <http://eudml.org/doc/215972>.

@article{Vũ1993,
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.},
author = {Vũ, Phóng},
journal = {Studia Mathematica},
keywords = {asymptotic behavior of -semigroups; nonquasianalytic weight function; spectral synthesis; contraction semigroups; Tauberian Theorem; Beurling algebras with nonquasianalytic weight},
language = {eng},
number = {3},
pages = {229-241},
title = {Semigroups with nonquasianalytic growth},
url = {http://eudml.org/doc/215972},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Vũ, Phóng
TI - Semigroups with nonquasianalytic growth
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 3
SP - 229
EP - 241
AB - 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.
LA - eng
KW - asymptotic behavior of -semigroups; nonquasianalytic weight function; spectral synthesis; contraction semigroups; Tauberian Theorem; Beurling algebras with nonquasianalytic weight
UR - http://eudml.org/doc/215972
ER -

References

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  7. [7] Yu. I. Lyubich, V. I. Matsaev and G. M. Fel'dman, Representations with separable spectrum, Funct. Anal. Appl. 7 (1973), 129-136. 
  8. [8] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42. Zbl0639.34050
  9. [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin 1983. 
  10. [10] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Univ. Press, Oxford 1968. Zbl0165.15601
  11. [11] Vũ Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84. Zbl0770.47017
  12. [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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