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Extreme Positive Contractions on Finite Dimensional lp-Spaces

Published online by Cambridge University Press:  20 November 2018

Ryszard Grząślewicz
Ryszard Grząślewicz*
Affiliation:
Technical University of Wrocław, Wrocław, Poland
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 682 - 699
Copyright
Copyright © Canadian Mathematical Society 1985

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