Vol. 80, No. 2, 1979

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On common fixed point sets of commutative mappings

Teck Cheong Lim

Vol. 80 (1979), No. 2, 517–521

Let C be a compact convex subset of a locally convex topological vector space X. Anzai and Ishikawa recently proved that if T1,,Tn is a finite commutative family of continuous affine self-mappings of C, then F( i=1nλiTi) = i=1nF(Ti) for every λi such that 0 < λi < 1 and i=1nλi = 1, where F(T) denotes the fixed point set of T. It is natural to question whether the conclusion of their theorem is dependent on the topological properties of X, C and Ti—in this case, the linear topology, the compactness and the continuity. We shall see that this is not; the theorem can be formulated in an algebraic context.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 3 March 1978
Published: 1 February 1979
Teck Cheong Lim