Abstract
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of ell ^{1}. We also point out some inaccurate assertions appearing in the literature on this topic.
Highlights
This brief note was inspired by recent papers by Bauschke et al [2], and Ryu [26]
Let (X, d) be a metric space and let T be a nonexpansive map of X into itself, that is, d(T x, T y) ≤ d(x, y) for all x, y ∈ X
(v), k=0 where πU is the projection onto the subspace of U -invariant vectors. It is stated in [2, Fact 1.1] and in [26] that for fixed-point free nonexpansive maps of Hilbert spaces the orbit must diverge in the sense that T nx → ∞ as n → ∞
Summary
Let (X, d) be a metric space and let T be a nonexpansive map of X into itself, that is, d(T x, T y) ≤ d(x, y) for all x, y ∈ X. In this paper we are mostly interested in the complementary case; when no fixed point exists. The case when X is a Banach space has been especially studied, see [1,2,13,16,17,18,20,22,23,26] and references therein. It should be pointed out that a significant special case was considered in the 1930s: the mean ergodic theorem of von Neumann and Carleman, especially in the generality of F. The iterates T n0 converge in the following sense: 0123456789().: V,-vol
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