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Equivalence of K-irreducibility concepts. (English) Zbl 0558.47031
The equivalence of various concepts of irreducibility of positive operators in partially ordered Banach spaces introduced by G. Frobenius (Fr), H. Geiringer (Ge), König (Ko), H. H. Schaefer (Sc), I. Sawashima (Sa), J. S. Vandergraft (VS), V. Ya. Stecenko (St) and I. Marek and K. Žitný (MZ) is analyzed. All the concepts considered are equivalent if the dimension of the spaces under consideration is at least two. In one-dimensional spaces these concepts split into two classes - the criterion being a classification of the zero map as reducible ((MZ), (Sa), (Sc)) or irreducible ((Fr), (Ge), (Ko), (St), (VS)), respectively.

47B60 Linear operators on ordered spaces
46B42 Banach lattices
46A40 Ordered topological linear spaces, vector lattices
15B48 Positive matrices and their generalizations; cones of matrices
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