# Semigroups with nonquasianalytic growth

Studia Mathematica (1993)

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

## Access Full Article

top## Abstract

top## How to cite

topVũ, 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

top- [1] G. R. Allan and T. J. Ransford, Power-dominated elements in a Banach algebra, Studia Math. 94 (1989), 63-79. Zbl0705.46021
- [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852. Zbl0652.47022
- [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. Zbl0088.32101
- [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. Zbl0611.47005
- [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. Zbl0639.34050
- [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. Zbl0165.15601
- [11] Vũ Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84. Zbl0770.47017
- [12] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semidynamical systems, J. Differential Equations, to appear.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.