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Almost periodic and strongly stable semigroups of operators

Vũ Phóng

Banach Center Publications (1997)

  • Volume: 38, Issue: 1, page 401-426
  • ISSN: 0137-6934

Abstract

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This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators T n : n = 0 , 1 , . . . on a Banach space X (discrete one-parameter semigroups), one-parameter C 0 -semigroups T ( t ) : t ≥ 0 on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for C 0 -semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and C 0 -semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for C 0 -semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.

How to cite

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Phóng, Vũ. "Almost periodic and strongly stable semigroups of operators." Banach Center Publications 38.1 (1997): 401-426. <http://eudml.org/doc/208644>.

@article{Phóng1997,
abstract = {This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators $\{T^\{n\}: n = 0,1,... \}$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_0$-semigroups $\{T(t): t ≥ 0\}$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for $C_0$-semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and $C_0$-semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for $C_0$-semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.},
author = {Phóng, Vũ},
journal = {Banach Center Publications},
keywords = {Abelian semigroup; representation; -semigroup; stability; asymptotics; almost periodicity},
language = {eng},
number = {1},
pages = {401-426},
title = {Almost periodic and strongly stable semigroups of operators},
url = {http://eudml.org/doc/208644},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Phóng, Vũ
TI - Almost periodic and strongly stable semigroups of operators
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 401
EP - 426
AB - This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators ${T^{n}: n = 0,1,... }$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_0$-semigroups ${T(t): t ≥ 0}$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded linear operators T(s): s ∈ S is called a representation of S in B(X) if: (i) T(s+t) = T(s)T(t); (ii) For every x ∈ X, s ↦ T(s)x is a continuous mapping from S to X. The central result which will be discussed in this article is a spectral criterion for almost periodicity of semigroups, obtained by Lyubich and the author [40] for uniformly continuous representations of arbitrary topological abelian semigroups (thus including the case of single bounded operators and several commuting bounded operators), and for $C_0$-semigroups [41], and by Batty and the author [9] for arbitrary strongly continuous representations of suitable locally compact abelian semigroups. An immediate consequence of this result is a Stability Theorem, obtained, for single operators and $C_0$-semigroups, also by Arendt and Batty [1] independently. The proof in [1] uses a Tauberian theorem for the Laplace-Stieltjes transforms and transfinite induction. Methods of this type can also be used to prove the almost periodicity result for $C_0$-semigroups [8], but seem not suitable for commuting semigroups, and will not be discussed in this article. We also refer the reader to a recent survey article of Batty [6], where some developments are described which are not included here.
LA - eng
KW - Abelian semigroup; representation; -semigroup; stability; asymptotics; almost periodicity
UR - http://eudml.org/doc/208644
ER -

References

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  1. [1] 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
  2. [2] R. Arens, Inverse-producing extensions of normed algebras, Trans. Amer. Math. Soc. 88 (1958) 536-548. Zbl0082.11105
  3. [3] W. B. Arveson, On groups of automorphisms of operator algebras, J. Funct. Anal. 15 (1974), 217-243. Zbl0296.46064
  4. [4] J. B. Baillon and P. Clément, Examples of unbounded imaginary powers of operators, ibid. 100 (1991), 419-434. 
  5. [5] W. Bartoszek, Asymptotic periodicity of the iterates of positive contractions on Banach lattices, Studia Math. 91 (1988), 179-188. Zbl0675.47025
  6. [6] C. J. K. Batty, Asymptotic behaviour of semigroups of operators in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 35-52. Zbl0818.47034
  7. [7] C. J. K. Batty, Z. Brzeźniak and D. Greenfield, A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum, Studia Math. 121 (1996), 167-183. Zbl0862.47020
  8. [8] C. J. K. Batty and Vũ Quôc Phóng, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805-818. 
  9. [9] C. J. K. Batty and Vũ Quôc Phóng, Stability of strongly continuous representations of abelian semigroups, Math. Z. 209 (1992), 75-88. 
  10. [10] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North- Hollland, Amsterdam, 1988. Zbl0663.47002
  11. [11] P. R. Chernoff, Two counterexamples in semigroup theory on Hilbert space, Proc. Amer. Math. Soc. 56 (1976), 253-255. Zbl0303.47030
  12. [12] I. Colojoarǎ and C. Foiaş, Theory of Generalized Spectral Operators, Gordon & Breach, New York, 1968. Zbl0189.44201
  13. [13] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, Berlin, 1994. Zbl0811.47034
  14. [14] R. deLaubenfels and Vũ Quôc Phóng, Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems, Proc. Amer. Math. Soc., to appear. Zbl0860.47025
  15. [15] R. deLaubenfels and Vũ Quôc Phóng, The discrete Hille-Yosida space, stability of individual orbits, and invariant subspaces, preprint. Zbl0959.47026
  16. [16] R. G. Douglas, On extending commutative semigroups of isometries, Bull. London Math. Soc. 1 (1969), 157-159. Zbl0187.06501
  17. [17] I. Erdelyi and S. W. Wang, A Local Spectral Theory for Closed Operators, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1985. Zbl0577.47035
  18. [18] J. Esterle, E. Strouse et F. Zouakia, Stabilité asymptotique de certains semigroupes d'opérateurs, J. Operator Theory 28 (1992), 203-227. 
  19. [19] S. R. Foguel, A counterexample to a problem of Sz.-Nagy, Proc. Amer. Math. Soc. 15 (1964), 788-790. Zbl0124.06602
  20. [20] I. Gelfand, Zur Theorie der Charaktere der abelschen topologischen Gruppen, Mat. Sb. 9 (51) (1941), 49-50. Zbl67.0407.02
  21. [21] P. R. Halmos, On Foguel's answer to Nagy's question, Proc. Amer. Math. Soc. 15 (1964), 791-793. Zbl0123.09701
  22. [22] P. R. Halmos, A Hilbert Space Problem Book, Springer, Berlin, 1982. 
  23. [23] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, R.I., 1957. 
  24. [24] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328. Zbl0611.47005
  25. [25] L. Kérchy, Unitary asymptotes of Hilbert space operators, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 191-201. Zbl0807.47005
  26. [26] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. 
  27. [27] A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751-764. Zbl0564.47015
  28. [28] Yu. I. Lyubich, On the spectrum of a representation of an abelian topological group, Dokl. Akad. Nauk SSSR 12 (1971), 1482-1486 (in Russian). Zbl0235.22008
  29. [29] Yu. I. Lyubich, V. I. Matsaev and G. M. Fel'dman, Representations with separable spectrum, Funct. Anal. Appl. 7 (1973), 129-136. 
  30. [30] Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birk- häuser, Basel, 1988. 
  31. [31] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42. Zbl0639.34050
  32. [32] A. McIntosh and A. Yagi, Operators of type ω without a bounded H ∞ -functional calculus, in: Miniconference on Operators in Analysis, Proc. Centre Math. Anal., ANU, Canberra (1989) 24, 159-172. 
  33. [33] M. Miklavčič, Asymptotic periodicity of the iterates of positivity preserving operators, Trans. Amer. Math. Soc. 307 (1988), 469-479. Zbl0654.46014
  34. [34] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. Zbl0201.45003
  35. [35] B. Sz.-Nagy, Completely continuous operators with uniformly bounded iterates, Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 (1959), 89-93. Zbl0086.31802
  36. [36] E. W. Packel, A semi-group analogue of Foguel's counterexample, Proc. Amer. Math. Soc. 21 (1969), 240-244. Zbl0175.13802
  37. [37] M. Putinar, Hyponormal operators are subscalar, J. Operator Theory 12 (1984), 385-395. Zbl0573.47016
  38. [38] C. R. Putnam, Hyponormal contractions and strong power convergence, Pacific J. Math. 57 (1975), 531-538. Zbl0321.47018
  39. [39] G. M. Sklyar and V. Ya. Shirman, On the asymptotic stability of linear differential equations in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 37 (1982), 127-132. Zbl0521.34063
  40. [40] Vũ Quôc Phóng and Yu I. Lyubich, A spectral criterion for asymptotic almost periodicity of uniformly continuous representations of abelian semigroups, J. Soviet Math. 51 (1990), 1263-1266. Originally published in Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 50 (1988), 38-43 (in Russian). 
  41. [41] Vũ Quôc Phóng and Yu I. Lyubich, A spectral criterion for almost periodicity of one-parameter semigroups, J. Soviet Math. 48 (1990), 644-647. Originally published in Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 47 (1987), 36-41 (in Russian). 
  42. [42] Vũ Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal. 103 (1992), 74-84. Zbl0770.47017
  43. [43] Vũ Quôc Phóng, A short proof of the Y. Katznelson's and L. Tzafriri's theorem, Proc. Amer. Math. Soc. 115 (1992), 1023-1024. Zbl0781.47003
  44. [44] Vũ Quôc Phóng, On the spectrum, complete trajectories and asymptotic stability of linear semidynamical systems, J. Differential Equations 105 (1993), 30-45. 
  45. [45] Vũ Quôc Phóng, Asymptotic almost periodicity and compactifying representations of semigroups, Ukrain. Math. J. 38 (1986), 576-579. Zbl0623.22005
  46. [46] Vũ Quôc Phóng, Stability of C 0 -semigroups commuting with a compact operator, Proc. Amer. Math. Soc., to appear. 
  47. [47] Vũ Quôc Phóng and F. Y. Yao, On similarity to contraction semigroups on Hilbert space, Semigroup Forum, to appear. 
  48. [48] J. Wermer, Banach Algebras and Several Complex Variables, Springer, New York, 1976. 
  49. [49] W. Żelazko, On a certain class of non-removable ideals in Banach algebras, Studia Math. 44 (1972), 87-92. Zbl0213.40603
  50. [50] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 369-385. Zbl0822.47005

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