Equicontinuous families of operators generating mean periodic maps

Valentina Casarino

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 3, page 141-171
  • ISSN: 1120-6330

Abstract

top
The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group U or an equicontinuous cosine function C forces the spectral structure of the infinitesimal generator of U or C . In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.

How to cite

top

Casarino, Valentina. "Equicontinuous families of operators generating mean periodic maps." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.3 (1999): 141-171. <http://eudml.org/doc/252315>.

@article{Casarino1999,
abstract = {The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group \( U \) or an equicontinuous cosine function \( C \) forces the spectral structure of the infinitesimal generator of \( U \) or \( C \). In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.},
author = {Casarino, Valentina},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Mean periodicity; Equicontinuous groups; Schwartz spectrum; mean periodicity; equicontinuous groups},
language = {eng},
month = {9},
number = {3},
pages = {141-171},
publisher = {Accademia Nazionale dei Lincei},
title = {Equicontinuous families of operators generating mean periodic maps},
url = {http://eudml.org/doc/252315},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Casarino, Valentina
TI - Equicontinuous families of operators generating mean periodic maps
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/9//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 3
SP - 141
EP - 171
AB - The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group \( U \) or an equicontinuous cosine function \( C \) forces the spectral structure of the infinitesimal generator of \( U \) or \( C \). In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.
LA - eng
KW - Mean periodicity; Equicontinuous groups; Schwartz spectrum; mean periodicity; equicontinuous groups
UR - http://eudml.org/doc/252315
ER -

References

top
  1. Arendt, W. - Batty, C.J.K., Almost periodic solutions of first and second order Cauchy problems. J. Diff. Eq., 137, 1997, 363-383. Zbl0879.34046MR1456602DOI10.1006/jdeq.1997.3266
  2. Arendt, W. - Grabosch, A. - Greiner, G. - Groh, U. - Lotz, H.P. - Moustakas, U. - Nagel, R. - Neubrander, F. - Schlotterbeck, U., One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, n. 1184, Springer-Verlag, Berlin - Heidelberg - New York - Tokyo1986. MR839450
  3. Beurling, A., A theorem on functions defined on semigroups. Math. Scand., 1, 1953, 127-130. Zbl0050.33701MR57467
  4. Bohr, H., Almost periodic functions. Chelsea, New York1947. Zbl0278.42019MR20163
  5. Carleman, T., L’integrale de Fourier et questions qui s’y rattachent. Almqvist and Wiksell, Uppsala1944. Zbl0060.25504MR14165
  6. Casarino, V., Quasi-periodicità e uniforme continuità di semigruppi e funzioni coseno: criteri spettrali. Doctoral Dissertation, Politecnico di Torino1998. 
  7. Casarino, V., Spectral properties of weakly almost periodic cosine functions. Rend. Mat. Acc. Lincei, s. 9, vol. 9, 1998, 177-211. Zbl0944.47027MR1683008
  8. Delsarte, J., Les fonctions moyenne-périodiques. Journal de Mathématiques pures et appliquées, 14, 1935, 403-453. Zbl0013.25405JFM61.1185.02
  9. Fattorini, H., Ordinary differential equations in linear topological spaces I. J. Differential Equations, 5, 1969, 72-105. Zbl0175.15101MR277860
  10. Fattorini, H., Ordinary differential equations in linear topological spaces II. J. Differential Equations, 6, 1969, 50-70. Zbl0181.42801MR277861
  11. Kahane, J.P., Sur quelques problèmes d’unicité et de prolongement, relatifs aux fonctions approchables par des sommes d’exponentielles. Annales de l’Institut Fourier, 5, 1954, 39-130. Zbl0064.35903MR75350
  12. Kahane, J.P., Sur les fonctions moyenne-périodiques bornées. Annales de l’Institut Fourier, 7, 1957, 293-314. Zbl0083.34401MR102702
  13. Kahane, J.P., Lectures on mean periodic functions. Tata Institut for fundamental Research, Bombay1958. Zbl0099.32301
  14. Katznelson, Y., An introduction to Harmonic Analysis. John Wiley & Sons, New York - London - Sydney - Toronto1968. Zbl0169.17902MR248482
  15. Kelley, J.L. - Namioka, I., Linear topological spaces. Springer-Verlag, New York - Heidelberg - Berlin1963. Zbl0318.46001MR166578
  16. Koosis, P., On functions which are mean periodic on a half-line. Comm. Pure and Appl. Math., 10, 1957, 133-149. Zbl0077.10302MR89297
  17. Nagy, B., On cosine operator functions in Banach spaces. Acta Sci. Math. (Szeged), 36, 1974, 281-290. Zbl0273.47008MR374995
  18. van Neerven, J.M.A.M., The adjoint of a semigroup of linear operators. Springer-Verlag, Berlin - Heidelberg - New York - Tokyo1992. Zbl0780.47026MR1222650
  19. van Neerven, J.M.A.M., The Asymptotic Behaviour of Semigroups of Linear Operators. Birkhäuser Verlag, Basel1996. Zbl0905.47001MR1409370DOI10.1007/978-3-0348-9206-3
  20. Pazy, A., Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York - Berlin - Heidelberg - Tokyo1983. Zbl0516.47023MR710486DOI10.1007/978-1-4612-5561-1
  21. Schwartz, L., Theorie générale des fonctions moyenne-périodiques. Annals of Mathematics, 48, 1947, 857-929. Zbl0030.15004MR23948
  22. Sova, M., Cosine operator functions. Rozprawy Matematyczne, XLIX, Warszawa1966. Zbl0156.15404MR193525
  23. Vesentini, E., Introduction to continuous semigroups. Scuola Normale Superiore, Pisa1996. Zbl1068.47001MR1736550
  24. Vesentini, E., Spectral properties of weakly asymptotically almost periodic semigroups. Advances in Math., 128, 1997, 217-241. Zbl0882.22002MR1454398DOI10.1006/aima.1997.1613
  25. Yosida, K., Functional Analysis. Springer Verlag, Berlin - Heidelberg - New York1968. Zbl0435.46002MR239384

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.