Download PDF - A note on convergence of semigroupsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

A note on convergence of semigroups

Authors 1

Affiliations

  1. Chair of Mathematics, Department of Electrical Engineering, Lublin Technical University, Nadbystrzycka 38a, 20-618 Lublin, Poland

Abstract

Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.

Keywords

ENG
semigroup, approximation, differentiable semigroup, resolvent, asymptotic behaviour, mean ergodic theorems

Bibliography

  1. N. H. Abdelazis, A note on convergence of linear semigroups of class (1.A), Hokkaido Math. J. 18 (1989), 513-521.
  2. N. H. Abdelazis, On approximation by discrete semigroups, J. Approx. Theory 73 (1993), 253-269.
  3. N. H. Abdelazis and P. R. Chernoff, Continuous and discrete semigroup approximations with applications to the Cauchy problems, J. Operator Theory 32 (1994), 331-352.
  4. W. Arendt, Vector-valued Laplace transforms and Cauchy problem, Israel J. Math. 59 (1987), 321-352.
  5. W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.
  6. C. J. K. Batty, Some Tauberian theorems related to operator theory, in: Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 21-34.
  7. C. J. K. Batty, Asymptotic behaviour of semigroups, in: Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 35-52.
  8. B. Bäumer and F. Neubrander, Laplace transform methods for evolution equations, Confer. Sem. Mat. Univ. Bari 1994, 27-60.
  9. A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum 49 (1994), 303-327.
  10. A. Bobrowski, Examples of a pointwise convergence of semigroups, Ann. Univ. Mariae Curie-Skłodowska Sect. A 49 (1995), 15-33.
  11. A. Bobrowski, Integrated semigroups and the Trotter-Kato theorem, Bull. Polish Acad. Sci. Math. 41 (1994), 297-304.
  12. A. Bobrowski, On the generation of non-continuous semigroups, Semigroup Forum 54 (1997), 237-252.
  13. A. Bobrowski, On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math. 124 (1997), 281-290.
  14. A. Bobrowski, On approximation of (1.A) semigroups by discrete semigroups, Bull. Polish Acad. Sci. Math. 46 (1998), 141-154.
  15. S. Busenberg and B. Wu, Convergence theorems for integrated semigroups, Differential Integral Equations 5 (1992), 509-520.
  16. J. T. Cannon, Convergence criteria for a sequence of semi-groups, Appl. Anal. 5 (1975), 23-31.
  17. P. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-205.
  18. P. C. Curtis and A. Figà-Talamanca, Factorization theorems for Banach algebras, in: Function Algebras, F. T. Birtel (ed.), Scott and Foresman, Chicago, 1966.
  19. G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola Norm. Sup. Pisa 14 (1987), 285-344.
  20. E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980.
  21. B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbation of differentiable semigroups, Semigroup Forum 54 (1997), 100-111.
  22. N. Dunford, Spectral theory, I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217.
  23. S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley Ser. Probab. Math. Statist., Wiley, New York, 1986.
  24. A. Favini, J. A. Goldstein and S. Romanelli, Analytic semigroups on Lwp and on Lp(0,1) generated by classes of second order differential operators, preprint, 1997.
  25. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monographs, 1985.
  26. J. A. Goldstein, C. Radin and R. E. Showalter, Convergence rates of ergodic limits for semigroups and cosine operator functions, Semigroup Forum 16 (1978), 89-95.
  27. E. Görlich and D. Pontzen, Approximation of operator semigroups of Oharu's class Ck, Tôhoku Math. J. (2) 34 (1982), 539-552.
  28. S. L. Gulik, T. S. Liu and A. C. M. van Rooij, Group algebra modules II, Canad. J. Math. 19 (1967), 151-173.
  29. B. Hennig and F. Neubrander, On representations, inversions and approximations of Laplace transform in Banach spaces, Appl. Anal. 49 (1993), 151-170.
  30. E. Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964) 147-155.
  31. E. Hille, On the differentiability of semigroups of operators, Acta Sci. Math. (Szeged) 12 (1950), 19-24.
  32. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, rev. ed., Amer. Math. Soc. Collloq. Publ. 31, Providence, R.I., 1957.
  33. K. Ito and H. P. Mc Kean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965.
  34. T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966.
  35. J. Kisyński, A proof of the Trotter-Kato theorem on approximation of semigroups, Colloq. Math. 18 (1967), 181-184.
  36. J. Kisyński, Semigroups of operators and some of their applications to partial differential equations, in: Control Theory and Topics in Functional Analysis, Vol. 3, IAEA, Vienna, 1978, 305-405.
  37. T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal. 3 (1969), 354-375.
  38. T. G. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 23-32.
  39. A. Lasota and R. Rudnicki, Asymptotic behaviour of semigroups of positive operators on C(X), Bull. Polish Acad. Sci. Math. 36 (1988), 151-159.
  40. C. Lizama, On the convergence and approximation of integrated semigroups, J. Math. Anal. Appl. 181, (1994), 89-103.
  41. G. Lumer, Solutions généralisées et semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I 310 (1990), 557-582.
  42. G. Lumer, A (very) direct approach to locally Lipschitz continuous integrated semigroups and some related new results oriented towards applications, via generalized solutions, in: LSU Seminar Notes in Functional Analysis and PDEs, 1990-1991, Louisiana State Univ., Baton Rouge, 1991, 88-107.
  43. G. Lumer, Evolution equations: Solutions for irregular evolution problems via generalized solutions and generalized initial values. Applications to periodic shocks models, Ann. Univ. Sarav. Ser. Math. 5 (1994), no. 1, 1-102.
  44. Yu. I. Lyubich and V u Quôc Phóng, Asymptotic stability of linear differential equations on Banach spaces, Studia Math. 88 (1988), 37-42.
  45. R. Nagel, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986.
  46. J. Neveu, Théorie des semi-groupes de Markov, Univ. Calif. Publ. Statist. 2 (1958), 319-394.
  47. A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1960), 1131-1141.
  48. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.
  49. R. S. Phillips, An inversion formula for the Laplace transform and semigroups of linear operators, Ann. of Math. 59 (1954), 325-356.
  50. M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum 51 (1995), 343-346.
  51. G. M. Sklyar and V. Ya. Shirman, On the asymptotic stability of a linear differential equation in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 37 (1982), 127-132 (in Russian).
  52. M. Slemrod, Asymptotic behaviour of C₀ semi-groups as determined by the spectrum of the generator, Indiana Univ. Math. J. 25 (1976), 783-792.
  53. T. Takahashi and S. Oharu, Approximation of operator semigroups in a Banach space, Tôhoku Math. J. 24 (1972), 505-528.
  54. K. Yosida, On the differentiability of semigroups of linear operators, Proc. Japan Acad. 34 (1958), 337-340.
  55. K. Yosida, Functional Analysis, Springer, Berlin, 1968.
Annales Polonici Mathematici
1998/69/2
Export to PBN, Opens in new tab
Pages:
107-127
Main language of publication
English
Received
1997-04-28
Published
1998
Exact and natural sciences