Extreme Positive Contractions on Finite Dimensional lp -Spaces

Grząślewicz, Ryszard

Canadian journal of mathematics, Tome 37 (1985) no. 4, pp. 682-699

Résumé

In this paper we give a characterization of the extreme positive contractions on finite dimensional lp -spaces for 1 < p < ∞. This is related to the characterization of the extreme doubly stochastic operators. In Section 2 we present the basic properties of the facial structure of the set of doubly stochastic n × m matrices. In Section 3 we use these facts for description of the facial structure of the set of positive contractions on finite dimensional lp -space. Next is considered stability of the positive part of the unit ball of operators (Section 5). In Section 7 we prove that extreme positive contractions on are strongly exposed. 1. Terminology and notation. Let (X, , m) be a a-finite measure space. As usual, we denote by LP(m), 1 < p < ∞, the Banach lattice of all p-summable real-valued functions on X with standard norm and order. If X = {1, 2 , ..., n) n < ∞, and m is a counting measure we write instead of LP(m). If X = [0, 1] and m is Lebesgue measure we write briefly LP .

Détail

DOI : 10.4153/CJM-1985-036-4

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