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Proceedings of the American Mathematical Society

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On commuting families of nonexpansive operators

Author: Gideon Schechtman
Journal: Proc. Amer. Math. Soc. 84 (1982), 373-376
MSC: Primary 47H09; Secondary 47H10
MathSciNet review: 640234
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Abstract: For each $ n$, $ 1 \leqslant n \leqslant \infty $, there exist a weakly compact, convex subset $ W$ of $ {L_1}$ and a family $ \{ {T_i}\} _{i = 1}^n(\{ {T_i}\} _{i = 1}^\infty {\text{ in case }}n = \infty )$ of nonexpansive operators of $ W$ into $ W$ such that there is no common fixed point for the whole family but any (finite) strict subfamily admits a common fixed point.

References [Enhancements On Off] (What's this?)

  • [1] D. E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423-424. MR 612733 (82j:47070)
  • [2] N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958.

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Keywords: Common fixed point, nonexpansive operators, weakly compact sets, convex sets
Article copyright: © Copyright 1982 American Mathematical Society

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