The aim of this paper is to give a complete proof of the necessity of the maximality conditions for both the analytic and the topological Krein-Rutman theorems in case of non-convex cone. In addition, we demonstrate the converse of Theorem 1, that is to say the converse to the K.-R. theorems, namely that any convex cone which satisfies the maximality conditions implies the existence of an analytic extreme point. In order to reach this goal, we study the expression of a closed and convex cone by means of a univalent function. We show that all the characters of the Krein-Rutman theorems can be established in this framework.
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A sufficient condition for the existence of an order preserving maximum of a function with respect to a cone is stated. The latter property is discussed and shown to be related to the notion of a weakly compact operator. In the last part of the paper, we present several sufficient conditions for the existence of an order preserving maximum of a function with respect to a cone in a functional analysis context. This approach was first suggested in [19] but the results and the main ideas developed in this paper are new.
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With the aim to generalize the standard results in finite dimensions, we define and study the concept of an A-range operator in a locally convex Hausdorff topological linear space. We examine in particular the topological and the norm properties of this operator and compare the two situations. The definition of the A-range is motivated by the property of the range of a range operator of being contained in the range of the operator.
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