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On latticially nonatomic subdifferentials. (English. Russian original) Zbl 0862.46011

Sib. Math. J. 35, No. 4, 760-765 (1994); translation from Sib. Mat. Zh. 35, No. 4, 853-859 (1994).
In [Interuniv. thematic Work Collect., No. 2, Yaroslavl 1978, 132-147 (1978; Zbl 0455.46025)] G. Ya. Lozanovskij characterized Banach \(K_\sigma\)-spaces \((E,|\cdot|)\) whose conjugate spaces \(E^*\) are nonatomic. In [Indag. Math., New Ser. 1, No. 3, 391-395 (1990; Zbl 0731.46008)] B. de Pagter and W. Wnuk extended the Lozanovskij result to the case of Banach lattices. In the present article we study operators that act from a vector lattice into an arbitrary \(K\)-space and establish criteria for latticial nonatomicity of a subdifferential or, in other words, the absence of nonzero lattice homomorphisms in the subdifferential under study. With the help of the criteria, we characterize a Banach-Kantorovich lattice whose conjugate space is latticially nonatomic. The obtained results generalize the criteria of Lozanovskij and de Pagter and Wnuk for nonatomicity of the conjugate of a Banach lattice.

MSC:

46B42 Banach lattices
46B40 Ordered normed spaces
47B60 Linear operators on ordered spaces
46G05 Derivatives of functions in infinite-dimensional spaces
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[1] G. Ya. Lozanovskii, ”Discrete functionals in Marcinkiewicz and Orlicz spaces,” Studies in the Theory of Functions of Several Real Variables [in Russian], Yaroslav. Univ., Yaroslavl’, 1978, No. 2, 132–147.
[2] B. Pagter and de W. Wnuk, ”Some remarks on Banach lattices with nonatomic duals,” Indag. Math. (N.S.),1, No. 3, 391–394 (1990). · Zbl 0731.46008
[3] A. G. Kusraev and S. S. Kutateladze, Subdifferentials: Theory and Applications [in Russian], Nauka, Novosibirsk (1992). · Zbl 0760.49012
[4] A. G. Kusraev, Vector Duality and Its Applications [in Russian], Novosibirsk, Nauka (1985). · Zbl 0616.49010
[5] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces [in Russian], Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1961). · Zbl 0101.08501
[6] A. V. Bukhvalov, V. B. Korotkov, A. G. Kusraev, et al., Vector Lattices and Integral Operators [in Russian], Nauka, Novosibirsk (1992). · Zbl 0752.46001
[7] A. G. Kusraev and S. S. Kutateladze, Nonstandard Methods of Analysis [in Russian], Nauka, Novosibirsk (1990). · Zbl 0718.03046
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