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Several order projectors generated by ideals of a vector lattice. (English. Russian original) Zbl 0877.47021

Sib. Math. J. 36, No. 6, 1164-1170 (1995); translation from Sib. Mat. Zh. 36, No. 6, 1342-1349 (1995).
Let \(T\) be a positive operator from a vector lattice \(E\) into a Dedekind complete vector lattice \(F\). C. D. Aliprantis and O. Burkinshaw [Indag. Math. 45, No. 1, 1-6 (1983; Zbl 0508.47037)] proved that the order continuous component \(T_n\) (resp. sequentially order continuous component \(T_{n\sigma})\) of the operator \(T\) can be computed by the formula \(T_n x=\inf \{\sup_\alpha Tx_\alpha: x_\alpha\uparrow x\}\) \((x\in E_+)\) [resp. \(T_{n\sigma}x=\inf \{\sup_n Tx_n: x_n\uparrow x\}\) \((x\in E_+)\)]. The paper under review deals with some generalizations and extensions of these formulas.

MSC:

47B65 Positive linear operators and order-bounded operators
46A40 Ordered topological linear spaces, vector lattices

Citations:

Zbl 0508.47037
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Full Text: DOI

References:

[1] E. V. Kolesnikov, ”Fragments of a positive operator,” in: Optimizatsiya [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1987, No. 40, pp. 140–146. · Zbl 0656.47014
[2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York (1985). (Pure Appl. Math.;119.) · Zbl 0608.47039
[3] B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces [in Russian], Fizmatgiz, Moscow (1961). · Zbl 0101.08501
[4] A. V. Bukhvalov, V. B. Korotkov, A. G. Kusraev, et al., Vector Lattices and Integral Operators [in Russian], Nauka, Novosibirsk (1992). · Zbl 0752.46001
[5] H. H. Shaefer and Zhang Xiao-Dong, ”Extension properties of order continuous functionals and applications to the theory of Banach lattices,” Indag. Math. (N. S.),5, No. 1, 107–118 (1994). · Zbl 0802.46032 · doi:10.1016/0019-3577(94)90037-X
[6] C. D. Aliprantis and O. Burkinshaw, ”On positive order continuous operators,” Indag. Math. (N. S.),45, No. 1, 1–6 (1983). · Zbl 0508.47037
[7] E. V. Kolesnikov, A. G. Kusraev, and S. A. Malyugin, On Dominated Operators [in Russian] [Preprint, No. 26], Inst. Mat. (Novosibirsk), Novosibirsk (1988).
[8] A. G. Kusraev and V. Z. Strizhevskii, ”Lattice-normed spaces and dominated operators,” in: Studies on Geometry and Mathematical Analysis [in Russian], Trudy Inst. Mat. Vol. 7 (Novosibirsk), Novosibirsk, 1987, pp. 132–157.
[9] A. G. Kusraev and S. A. Malyugin, ”On an order continuous component of a dominated operator,” Sibirsk. Mat. Zh.,31, No. 4, 127–139 (1987). · Zbl 0643.47043
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