Geodesic
Wold Decomposition in Banach Spaces
Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 894-904.

Voir la notice de l'article provenant de la source Math-Net.Ru

We propose a natural analog of the Wold decomposition in the case of a linear noninvertible isometry $V$ in a Banach space $X$. We obtain a criterion for the existence of such a decomposition. In a reflective space, this criterion is reduced to the existence of the linear projection $P\colon X\to V\!X$ with unit norm. Separately, we discuss the problem of the Wold decomposition for the isometry $V_\varphi$ induced by an epimorphism $\varphi$ of a compact set $H$ in the space of continuous functions $C(H)$. We present a detailed study of the mapping $z\to z^m$ of the circle $|z|=1$ with an integer $m\ge2$.
Keywords: Wold decomposition, linear noninvertible isometry, Banach space, reflexive space, unitary operator, completely nonunitary isometry, one-sided shift. 
@article{MZM_2007_82_6_a9,
     author = {A. V. Romanov},
     title = {Wold {Decomposition} in {Banach} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {894--904},
     publisher = {mathdoc},
     volume = {82},
     number = {6},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a9/}
}
TY  - JOUR
AU  - A. V. Romanov
TI  - Wold Decomposition in Banach Spaces
JO  - Matematičeskie zametki
PY  - 2007
SP  - 894
EP  - 904
VL  - 82
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a9/
LA  - ru
ID  - MZM_2007_82_6_a9
ER  - 
%0 Journal Article
%A A. V. Romanov
%T Wold Decomposition in Banach Spaces
%J Matematičeskie zametki
%D 2007
%P 894-904
%V 82
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a9/
%G ru
%F MZM_2007_82_6_a9
A. V. Romanov. Wold Decomposition in Banach Spaces. Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 894-904. http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a9/

[1] B. Sekefalvi-Nad, Ch. Foyash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[2] P. R. Khalmosh, Gilbertovo prostranstvo v zadachakh, Mir, M., 1970 | MR | Zbl

[3] N. K. Nikolskii, Lektsii ob operatore sdviga, Nauka, M., 1980 | MR | Zbl

[4] N. Danford, Dzh. T. Shvarts, Lineinye operatory, ch. 1. Obschaya teoriya, URSS, M., 2004 | MR | Zbl

[5] V. P. Odinets, M. Ya. Yakubson, Proektory i bazisy v normirovannykh prostranstvakh, URSS, M., 2004 | MR | Zbl

[6] R. Sato, “On abstract mean ergodic theorems”, Tohoku Math. J. (2), 30:4 (1978), 575–581 | DOI | MR | Zbl

[7] E. R. Lorch, “On a calculus of operators in reflexive vector spaces”, Trans. Amer. Math. Soc., 45:2 (1939), 217–234 | DOI | MR | Zbl

[8] A. Pelczynski, “Projections in certain Banach spaces”, Studia Math., 19:2 (1960), 209–228 | MR | Zbl

[9] R. M. Crownover, “Commutants of shifts on Banach spaces,”, Michigan Math. J., 19:3 (1972), 233–247 | DOI | MR | Zbl

[10] E. Michael, “A linear mapping between function spaces”, Proc. Amer. Math. Soc., 15:3 (1964), 407–409 | DOI | MR | Zbl

[11] S. P. Lloyd, “On extreme averaging operators”, Proc. Amer. Math. Soc., 14:2 (1963), 305–310 | DOI | MR | Zbl

[12] A. B. Katok, B. Khasselblat, Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999 | MR | Zbl