A note on convergence of semigroups

Adam Bobrowski

Annales Polonici Mathematici (1998)

  • Volume: 69, Issue: 2, page 107-127
  • ISSN: 0066-2216

Abstract

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

How to cite

top

Adam Bobrowski. "A note on convergence of semigroups." Annales Polonici Mathematici 69.2 (1998): 107-127. <http://eudml.org/doc/270524>.

@article{AdamBobrowski1998,
abstract = {Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.},
author = {Adam Bobrowski},
journal = {Annales Polonici Mathematici},
keywords = {semigroup; approximation; differentiable semigroup; resolvent; asymptotic behaviour; mean ergodic theorems; convergence of semigroups; holomorphic semigroup; Trotter-Kato-Neveu theorem; -semigroups; almost uniform convergence},
language = {eng},
number = {2},
pages = {107-127},
title = {A note on convergence of semigroups},
url = {http://eudml.org/doc/270524},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Adam Bobrowski
TI - A note on convergence of semigroups
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 2
SP - 107
EP - 127
AB - Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.
LA - eng
KW - semigroup; approximation; differentiable semigroup; resolvent; asymptotic behaviour; mean ergodic theorems; convergence of semigroups; holomorphic semigroup; Trotter-Kato-Neveu theorem; -semigroups; almost uniform convergence
UR - http://eudml.org/doc/270524
ER -

References

top
  1. [1] N. H. Abdelazis, A note on convergence of linear semigroups of class (1.A), Hokkaido Math. J. 18 (1989), 513-521. 
  2. [2] N. H. Abdelazis, On approximation by discrete semigroups, J. Approx. Theory 73 (1993), 253-269. 
  3. [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. [4] W. Arendt, Vector-valued Laplace transforms and Cauchy problem, Israel J. Math. 59 (1987), 321-352. 
  5. [5] 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
  6. [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. Zbl0810.40005
  7. [7] C. J. K. Batty, Asymptotic behaviour of semigroups, in: Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 35-52. Zbl0818.47034
  8. [8] B. Bäumer and F. Neubrander, Laplace transform methods for evolution equations, Confer. Sem. Mat. Univ. Bari 1994, 27-60. 
  9. [9] A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum 49 (1994), 303-327. Zbl0817.47047
  10. [10] A. Bobrowski, Examples of a pointwise convergence of semigroups, Ann. Univ. Mariae Curie-Skłodowska Sect. A 49 (1995), 15-33. 
  11. [11] A. Bobrowski, Integrated semigroups and the Trotter-Kato theorem, Bull. Polish Acad. Sci. Math. 41 (1994), 297-304. Zbl0824.47031
  12. [12] A. Bobrowski, On the generation of non-continuous semigroups, Semigroup Forum 54 (1997), 237-252. Zbl0873.47025
  13. [13] A. Bobrowski, On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math. 124 (1997), 281-290. Zbl0876.44001
  14. [14] A. Bobrowski, On approximation of (1.A) semigroups by discrete semigroups, Bull. Polish Acad. Sci. Math. 46 (1998), 141-154. Zbl0916.47030
  15. [15] S. Busenberg and B. Wu, Convergence theorems for integrated semigroups, Differential Integral Equations 5 (1992), 509-520. Zbl0786.47036
  16. [16] J. T. Cannon, Convergence criteria for a sequence of semi-groups, Appl. Anal. 5 (1975), 23-31. Zbl0366.47017
  17. [17] P. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-205. Zbl0085.10201
  18. [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. [19] G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola Norm. Sup. Pisa 14 (1987), 285-344. Zbl0652.34069
  20. [20] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
  21. [21] B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbation of differentiable semigroups, Semigroup Forum 54 (1997), 100-111. Zbl0867.47034
  22. [22] N. Dunford, Spectral theory, I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217. Zbl0063.01185
  23. [23] S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley Ser. Probab. Math. Statist., Wiley, New York, 1986. 
  24. [24] A. Favini, J. A. Goldstein and S. Romanelli, Analytic semigroups on L w p and on L p ( 0 , 1 ) generated by classes of second order differential operators, preprint, 1997. Zbl0939.47036
  25. [25] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monographs, 1985. 
  26. [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. Zbl0393.47004
  27. [27] E. Görlich and D. Pontzen, Approximation of operator semigroups of Oharu’s class C ( k ) , Tôhoku Math. J. (2) 34 (1982), 539-552. Zbl0498.47016
  28. [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. [29] B. Hennig and F. Neubrander, On representations, inversions and approximations of Laplace transform in Banach spaces, Appl. Anal. 49 (1993), 151-170. Zbl0791.44002
  30. [30] E. Hewitt, The ranges of certain convolution operators, Math. Scand. 15 (1964) 147-155. Zbl0135.36002
  31. [31] E. Hille, On the differentiability of semigroups of operators, Acta Sci. Math. (Szeged) 12 (1950), 19-24. Zbl0035.35802
  32. [32] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, rev. ed., Amer. Math. Soc. Collloq. Publ. 31, Providence, R.I., 1957. Zbl0078.10004
  33. [33] K. Ito and H. P. Mc Kean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965. 
  34. [34] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. 
  35. [35] J. Kisyński, A proof of the Trotter-Kato theorem on approximation of semigroups, Colloq. Math. 18 (1967), 181-184. Zbl0152.33905
  36. [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. [37] T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal. 3 (1969), 354-375. Zbl0174.18401
  38. [38] T. G. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 23-32. Zbl0194.44103
  39. [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. Zbl0676.47021
  40. [40] C. Lizama, On the convergence and approximation of integrated semigroups, J. Math. Anal. Appl. 181, (1994), 89-103. Zbl0815.47053
  41. [41] G. Lumer, Solutions généralisées et semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I 310 (1990), 557-582. 
  42. [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. [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. Zbl0813.34057
  44. [44] Yu. I. Lyubich and V u Quôc Phóng, Asymptotic stability of linear differential equations on Banach spaces, Studia Math. 88 (1988), 37-42. Zbl0639.34050
  45. [45] R. Nagel, One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986. Zbl0585.47030
  46. [46] J. Neveu, Théorie des semi-groupes de Markov, Univ. Calif. Publ. Statist. 2 (1958), 319-394. Zbl0173.45703
  47. [47] A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1960), 1131-1141. Zbl0162.45903
  48. [48] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. 
  49. [49] R. S. Phillips, An inversion formula for the Laplace transform and semigroups of linear operators, Ann. of Math. 59 (1954), 325-356. Zbl0059.10704
  50. [50] M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum 51 (1995), 343-346. Zbl0831.47030
  51. [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). Zbl0521.34063
  52. [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. Zbl0313.47026
  53. [53] T. Takahashi and S. Oharu, Approximation of operator semigroups in a Banach space, Tôhoku Math. J. 24 (1972), 505-528. Zbl0281.47024
  54. [54] K. Yosida, On the differentiability of semigroups of linear operators, Proc. Japan Acad. 34 (1958), 337-340. Zbl0083.11003
  55. [55] K. Yosida, Functional Analysis, Springer, Berlin, 1968. 

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.