Abstract
Let E be a Banach space partially ordered by a cone K. Let B be a closed linear operator in E with domain Γ(B). In this paper certain cones in the Banach space Γ(B) with norm ∥x∥Γ =∥x∥+∥Bx∥ are singled out for study; a number of their geometric properties are established under the assumption that the cone K in the space E has analogous properties.
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M. G. Krein and M. A. Rutman, “Linear operators which leave a cone in a Banach space invariant,” Uspekhi Matem. Nauk,3, No. 1, 3–95 (1948).
M. A. Krasnosel'skii, Positive Solutions of Operator Equations [in Russian], Moscow (1962).
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Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 63–70, 1969.
The author is indebted to M. A. Krasnosel'skii and V. Ya. Stetsenko for helpful discussions and for their interest in the results of the present paper.
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Sabirov, T. Geometric properties of some special cones in a Banach space. Mathematical Notes of the Academy of Sciences of the USSR 5, 40–44 (1969). https://doi.org/10.1007/BF01098714
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DOI: https://doi.org/10.1007/BF01098714