Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stability of individual elements under one-parameter semigroups


Authors: Charles J. K. Batty and Quôc Phóng Vù
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $ \vert\vert T(t)x\vert\vert \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 [Enhancements On Off] (What's this?)

  • [1] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852. MR 933321 (89g:47053)
  • [2] C. J. K. Batty, Tauberian theorems for the Laplace-Stieltjes transform, Trans. Amer. Math. Soc. 322 (1990), 783-804. MR 1013326 (91c:44001)
  • [3] Ph. Clément et al., One-parameter semigroups, North-Holland, Amsterdam, 1987. MR 915552 (89b:47058)
  • [4] K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63-97. MR 0131784 (24:A1632)
  • [5] B. de Pagter, A characterization of sun-reflexivity, Math. Ann. 283 (1989), 511-518. MR 985246 (90a:47102)
  • [6] N. Dunford and J. T. Schwartz, Linear operators. I, Wiley, New York, 1958.
  • [7] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc., Providence R.I., 1957. MR 0089373 (19:664d)
  • [8] A. E. Ingham, On Wiener's method in Tauberian theorems, Proc. London Math. Soc. (2) 38 (1935), 458-480.
  • [9] J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108-115. MR 684025 (84b:10063)
  • [10] V. I. Korobov and G. M. Sklyar, On the problem of strong stabilizability of contraction systems on Hilbert space, Differential Equations 20 (1984), 1320-1326. MR 773939 (86g:93052)
  • [11] U. Krengel, Ergodic theorems, De Gruyter, Berlin, 1985. MR 797411 (87i:28001)
  • [12] N. Levan and L. Rigby, Strong stabilizability of linear contractive control systems on Hilbert space, SIAM J. Control Optim. 17 (1979), 23-35. MR 516853 (80f:93061)
  • [13] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42. MR 932004 (89e:47062)
  • [14] R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer, Berlin, 1986. MR 839450 (88i:47022)
  • [15] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)
  • [16] R. E. O'Brien, Contraction semigroups, stabilization, and the mean ergodic theorem, Proc. Amer. Math. Soc. 71 (1978), 89-94. MR 495844 (80a:93140)
  • [17] Vũ Quôc Phóng, Représentations compactifiantes de semi-groupes, C.R. Acad. Sci. Paris Sér. A 305 (1987), 273-274. MR 907960 (88i:22002)
  • [18] -, Applications of Suskevic kernel to semigroup actions and representations, preprint, 1987.
  • [19] Vũ Quôc Phóng and Yu. I. Lyubich, A spectral criterion for almost periodicity of one-parameter semigroups, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 47 (1987), 36-41. (Russian). MR 916441 (89a:47067)
  • [20] M. Slemrod, A note on complete controllability and stabilizability of linear control systems in Hilbert space, SIAM J. Control Optim. 12 (1974), 500-508. MR 0353107 (50:5593)
  • [21] K. Yosida and S. Kakutani, Operator theoretical treatment of Markov processes and mean ergodic theorems, Ann. of Math. (2) 42 (1941), 188-228. MR 0003512 (2:230e)
  • [22] D. Zagier, Short proof of the prime number theorem, unpublished manuscript.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D03

Retrieve articles in all journals with MSC: 47D03


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1022866-5
Keywords: $ {C_0}$-semigroup, stability, residual spectrum, sun-reflexive, stabilization, almost periodic
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society