Some characterizations of amenable semigroups

AbstractWe first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

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Strong convergence of Browder’s type iterations for left amenable semigroups of Lipschitzian mappings in Banach spaces

Journal of Fixed Point Theory and Applications ◽

10.1007/s11784-008-0092-3 ◽

2008 ◽

Vol 5 (1) ◽

pp. 93-103 ◽

Cited By ~ 11

Author(s):

Shahram Saeidi

Keyword(s):

Strong Convergence ◽

Banach Spaces ◽

Lipschitzian Mappings ◽

Amenable Semigroups

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Amenability and invariant subspaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700019923 ◽

1974 ◽

Vol 18 (2) ◽

pp. 200-204 ◽

Cited By ~ 3

Author(s):

Anthony To-Ming Lau

Keyword(s):

Convex Subset ◽

Invariant Subspaces ◽

Complex Field ◽

Open Convex ◽

Amenable Semigroup ◽

Linear Action ◽

Banach Theorem ◽

Amenable Semigroups ◽

Open Convex Subset ◽

Invariant Element

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

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