## Stability of individual elements under one-parameter semigroups

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- by Charles J. K. Batty and Quôc Phóng Vù PDF
- Trans. Amer. Math. Soc.
**322**(1990), 805-818 Request permission

## Abstract:

Let $\{ T(t):t \geqslant 0\}$ be a ${C_0}$-semigroup on a Banach space $X$ with generator $A$, and let $x \in X$. If $\sigma (A) \cap i{\mathbf {R}}$ is empty and $t \mapsto T(t)x$ is uniformly continuous, then $||T(t)x|| \to 0$ as $t \to \infty$. If the semigroup is sun-reflexive, $\sigma (A) \cap i{\mathbf {R}}$ is countable, $P\sigma (A) \cap i{\mathbf {R}}$ is empty, and $t \mapsto T(t)x$ is uniformly weakly continuous, then $T(t)x \to 0$ weakly as $t \to \infty$. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.## References

- W. Arendt and C. J. K. Batty,
*Tauberian theorems and stability of one-parameter semigroups*, Trans. Amer. Math. Soc.**306**(1988), no. 2, 837–852. MR**933321**, DOI 10.1090/S0002-9947-1988-0933321-3 - C. J. K. Batty,
*Tauberian theorems for the Laplace-Stieltjes transform*, Trans. Amer. Math. Soc.**322**(1990), no. 2, 783–804. MR**1013326**, DOI 10.1090/S0002-9947-1990-1013326-6 - Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter,
*One-parameter semigroups*, CWI Monographs, vol. 5, North-Holland Publishing Co., Amsterdam, 1987. MR**915552** - K. de Leeuw and I. Glicksberg,
*Applications of almost periodic compactifications*, Acta Math.**105**(1961), 63–97. MR**131784**, DOI 10.1007/BF02559535 - B. de Pagter,
*A characterization of sun-reflexivity*, Math. Ann.**283**(1989), no. 3, 511–518. MR**985246**, DOI 10.1007/BF01442743
N. Dunford and J. T. Schwartz, - Einar Hille and Ralph S. Phillips,
*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR**0089373**
A. E. Ingham, - J. Korevaar,
*On Newman’s quick way to the prime number theorem*, Math. Intelligencer**4**(1982), no. 3, 108–115. MR**684025**, DOI 10.1007/BF03024240 - V. I. Korobov and G. M. Sklyar,
*On the question of the strong stabilizability of contracting systems in Hilbert space*, Differentsial′nye Uravneniya**20**(1984), no. 11, 1862–1869, 2019 (Russian). MR**773939** - Ulrich Krengel,
*Ergodic theorems*, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411**, DOI 10.1515/9783110844641 - N. Levan and L. Rigby,
*Strong stabilizability of linear contractive control systems on Hilbert space*, SIAM J. Control Optim.**17**(1979), no. 1, 23–35. MR**516853**, DOI 10.1137/0317003 - Yu. I. Lyubich and Vũ Quốc Phóng,
*Asymptotic stability of linear differential equations in Banach spaces*, Studia Math.**88**(1988), no. 1, 37–42. MR**932004**, DOI 10.4064/sm-88-1-37-42 - W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck,
*One-parameter semigroups of positive operators*, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR**839450**, DOI 10.1007/BFb0074922 - Béla Sz.-Nagy and Ciprian Foiaş,
*Harmonic analysis of operators on Hilbert space*, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR**0275190** - Robert E. O’Brien Jr.,
*Contraction semigroups, stabilization, and the mean ergodic theorem*, Proc. Amer. Math. Soc.**71**(1978), no. 1, 89–94. MR**495844**, DOI 10.1090/S0002-9939-1978-0495844-2 - Vũ Quốc Phóng,
*Représentations compactifiantes de semi-groupes*, C. R. Acad. Sci. Paris Sér. I Math.**305**(1987), no. 6, 273–274 (French, with English summary). MR**907960**
—, - Vu Kuok Fong and Yu. I. Lyubich,
*A spectral criterion for almost periodicity for one-parameter semigroups*, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen.**47**(1987), 36–41 (Russian); English transl., J. Soviet Math.**48**(1990), no. 6, 644–647. MR**916441**, DOI 10.1007/BF01094717 - Marshall Slemrod,
*A note on complete controllability and stabilizability for linear control systems in Hilbert space*, SIAM J. Control**12**(1974), 500–508. MR**0353107** - Kôsaku Yosida and Shizuo Kakutani,
*Operator-theoretical treatment of Markoff’s process and mean ergodic theorem*, Ann. of Math. (2)**42**(1941), 188–228. MR**3512**, DOI 10.2307/1968993
D. Zagier,

*Linear operators*. I, Wiley, New York, 1958.

*On Wiener’s method in Tauberian theorems*, Proc. London Math. Soc. (2)

**38**(1935), 458-480.

*Applications of Suskevic kernel to semigroup actions and representations*, preprint, 1987.

*Short proof of the prime number theorem*, unpublished manuscript.

## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**322**(1990), 805-818 - MSC: Primary 47D03
- DOI: https://doi.org/10.1090/S0002-9947-1990-1022866-5
- MathSciNet review: 1022866