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

Citations:

Zbl 0455.46025; Zbl 0731.46008
Full Text:

References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.