On uniform ergodic theorems for quadratic processes on $C^*$-algebras
Sbornik. Mathematics, Tome 191 (2000) no. 12, pp. 1891-1903 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate questions of the convergence in the uniform topology of Ceséaro time averages of stationary quantum quadratic processes defined on $C^*$-algebras. A necessary and sufficient condition is given for the convergence of averages in the uniform topology. In addition, the convergence of weighted averages in the uniform topology is investigated and it is proved that averages weighted by Besicovitch $\Phi$-functions converge.
@article{SM_2000_191_12_a6,
     author = {F. M. Mukhamedov},
     title = {On uniform ergodic theorems for quadratic processes on $C^*$-algebras},
     journal = {Sbornik. Mathematics},
     pages = {1891--1903},
     year = {2000},
     volume = {191},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_12_a6/}
}
TY  - JOUR
AU  - F. M. Mukhamedov
TI  - On uniform ergodic theorems for quadratic processes on $C^*$-algebras
JO  - Sbornik. Mathematics
PY  - 2000
SP  - 1891
EP  - 1903
VL  - 191
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2000_191_12_a6/
LA  - en
ID  - SM_2000_191_12_a6
ER  - 
%0 Journal Article
%A F. M. Mukhamedov
%T On uniform ergodic theorems for quadratic processes on $C^*$-algebras
%J Sbornik. Mathematics
%D 2000
%P 1891-1903
%V 191
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2000_191_12_a6/
%G en
%F SM_2000_191_12_a6
F. M. Mukhamedov. On uniform ergodic theorems for quadratic processes on $C^*$-algebras. Sbornik. Mathematics, Tome 191 (2000) no. 12, pp. 1891-1903. http://geodesic.mathdoc.fr/item/SM_2000_191_12_a6/

[1] Bernshtein S. N., “Reshenie odnoi matematicheskoi problemy, svyazannoi s teoriei nasledstvennosti”, Uch. zapiski NI kafedr Ukrainy, 1924, no. 1, 83–115

[2] Ulam S., Nereshennye matematicheskie zadachi, Nauka, M., 1964 | Zbl

[3] Kesten H., “Quadratic transformations: a model for population growth. I; II”, Adv. Appl. Prob., 2 (1970), 1–82 | DOI | MR | Zbl

[4] Lyubich Yu. I., “Osnovnye ponyatiya i teoremy evolyutsionnoi genetiki svobodnykh populyatsii”, UMN, 26:5 (1971), 51–116 | MR | Zbl

[5] Sarymsakov T. A., Ganikhodzhaev R. N., “Ergodicheskii printsip dlya kvadratichnykh stokhasticheskikh operatorov”, Izv. AN UzSSR. Ser. fiz.-matem. nauk, 1979, no. 6, 34–38 | MR

[6] Ganikhodzhaev N. N., Mukhamedov F. M., “O kvantovykh kvadratichnykh stokhasticheskikh protsessakh”, Dokl. AN RUz., 1997, no. 3, 13–16 | MR | Zbl

[7] Ganikhodzhaev N. N., Mukhamedov F. M., “O kvantovykh kvadratichnykh stokhasticheskikh protsessakh i nekotorye ergodicheskie teoremy dlya takikh protsessov”, Uzb. matem. zhurn., 1997, no. 3, 8–20 | MR | Zbl

[8] Sakai S., $C^*$-algebras and $W^*$-algebras, Springer-Verlag, Berlin, 1971 | MR | Zbl

[9] Lyubich Yu. I., Matematicheskie struktury v populyatsionnoi genetike, Naukova dumka, Kiev, 1983 | MR

[10] Janks R. D., “Quadratic differential systems for interactive population models”, J. Differential Equations, 5:3 (1969), 497–514 | DOI | MR

[11] Sarymsakov T. A., Ganikhodzhaev N. N., “Analiticheskie metody v teorii kvadratichnykh stokhasticheskikh operatorov”, Dokl. AN SSSR, 305:5 (1989), 1052–1056 | MR | Zbl

[12] Sarymsakov T. A., Ganikhodzhaev N. N., “Ob ergodicheskom printsipe dlya kvadratichnykh protsessov”, Dokl. AN SSSR, 316:6 (1991), 1315–1319 | MR | Zbl

[13] Kornfeld N. P., Sinai Ya. G., Fomin S. V., Ergodicheskaya teoriya, Nauka, M., 1980 | MR | Zbl

[14] Hensz E., “On some ergodic theorems for von Neumann algebras”, Probability theory on vector spaces. III, Proc. Conf. (Lublin/Pol. 1983), Lecture Notes in Math., 1080, 1984, 119–123 | MR | Zbl

[15] Litvinov S., “On weighted ergodic theorems in von Neumann algebras” (to appear)