On unrestricted products of (W) contractions

W. Bartoszek

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 2, page 163-170
  • ISSN: 0010-1354

Abstract

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Given a family of (W) contractions T 1 , . . . , T N on a reflexive Banach space X we discuss unrestricted sequences T r n ∘ . . . ∘ T r 1 ( x ) . We show that they converge weakly to a common fixed point, which depends only on x and not on the order of the operators T r n if and only if the weak operator closed semigroups generated by T 1 , . . . , T N are right amenable.

How to cite

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Bartoszek, W.. "On unrestricted products of (W) contractions." Colloquium Mathematicae 86.2 (2000): 163-170. <http://eudml.org/doc/210846>.

@article{Bartoszek2000,
abstract = {Given a family of (W) contractions $T_1, ..., T_N$ on a reflexive Banach space X we discuss unrestricted sequences $T_\{r_n\}∘...∘T_\{r_1\}(x)$. We show that they converge weakly to a common fixed point, which depends only on x and not on the order of the operators $T_\{r_n\}$ if and only if the weak operator closed semigroups generated by $T_1, ..., T_N$ are right amenable.},
author = {Bartoszek, W.},
journal = {Colloquium Mathematicae},
keywords = {weak convergence; unrestricted products; linear contraction; common fixed point},
language = {eng},
number = {2},
pages = {163-170},
title = {On unrestricted products of (W) contractions},
url = {http://eudml.org/doc/210846},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Bartoszek, W.
TI - On unrestricted products of (W) contractions
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 2
SP - 163
EP - 170
AB - Given a family of (W) contractions $T_1, ..., T_N$ on a reflexive Banach space X we discuss unrestricted sequences $T_{r_n}∘...∘T_{r_1}(x)$. We show that they converge weakly to a common fixed point, which depends only on x and not on the order of the operators $T_{r_n}$ if and only if the weak operator closed semigroups generated by $T_1, ..., T_N$ are right amenable.
LA - eng
KW - weak convergence; unrestricted products; linear contraction; common fixed point
UR - http://eudml.org/doc/210846
ER -

References

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  1. [AA] I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239-244. Zbl0143.16202
  2. [B] R. E. Bruck, Random products of contractions in metric and Banach spaces, J. Math. Anal. Appl. 88 (1982), 319-332. Zbl0512.47042
  3. [BA] H. H. Bauschke, A norm convergence result of random products of relaxed projections in Hilbert space, Trans. Amer. Math. Soc. 347 (1995), 1365-1373. Zbl0832.47055
  4. [DKLR] J. M. Dye, T. Kuczumow, P.-K. Lin and S. Reich, Convergence on unrestricted products of nonexpansive mappings in spaces with the Opial property, Nonlinear Anal. 26 (1996), 767-773. Zbl0866.47038
  5. [DLG] K. DeLeeuw and I. Glickberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63-97. 
  6. [D] J. Dye, A generalization of a theorem of Amemiya and Ando on the convergence of random products of contractions in Hilbert space, Integral Equations Oper. Theory 12 (1989), 155-162. Zbl0691.47012
  7. [DKR] J. Dye, M. A. Khamsi and S. Reich, Random products of contractions in Banach spaces, Trans. Amer. Math. Soc. 325 (1991), 87-99. Zbl0735.47001
  8. [DR] J. M. Dye and S. Reich, On the unrestricted iteration of projections in Hilbert space, J. Math. Anal. Appl. 156 (1991), 101-119. Zbl0731.65041
  9. [L] P.-K. Lin, Unrestricted products of contractions in Banach spaces, Nonlinear Anal. 24 (1995), 1103-1108. Zbl0870.47034
  10. [N] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 44 (1943), 401-485. Zbl0034.06102
  11. [R] S. Reich, The alternating algorithm of von Neumann in the Hilbert ball, Dynamic Systems Appl. 2 (1993), 21-26. Zbl0768.41032
  12. [RZ] S. Reich and A. J. Zaslavski, Convergence of generic infinite products of order-preserving mappings, Positivity 3 (1999), 1-21. Zbl0923.47032

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