Abstract
In this article we calculate the order projection in a space \(\mathcal{OA}_{r}(E,F)\) of all regular orthogonally additive operators from a vector lattice \(E\) to a Dedekind complete vector lattice \(F\), onto the band \(\{T\}^{\perp\perp}\) of \(\mathcal{OA}_{r}(E,F)\) which is generated by a disjointness preserving orthogonally additive operator \(T\colon E\to F\).
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Notes
\((C_{1})\) and \((C_{2})\) are called the Carathéodory conditions.
REFERENCES
N. Abasov, ‘‘Completely additive and \(C\)-compact operators in lattice-normed spaces,’’ Ann. Funct. Anal. 11, 914–928 (2020).
N. Abasov,‘‘On the sum of narrow orthogonally additive operators,’’ Russ. Math. 64 (7), 1–6 (2020).
N. Abasov and M. Pliev, ‘‘On extensions of some nonlinear maps in vector lattices,’’ J. Math. Anal. Appl. 455, 516–527 (2017).
N. Abasov and M. Pliev, ‘‘Disjointness-preserving orthogonally additive operators in vector lattices,’’ Banach J. Math. Anal. 12, 730–750 (2018).
C. D. Aliprantis and O. Burkinshaw, ‘‘The components of a positive operator,’’ Math. Z. 184, 245–257 (1983).
C. D. Aliprantis and O. Burkinshaw, Positive Operators (Springer, Dordrecht, 2006).
J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators (Cambridge Univ. Press, Cambridge, 1990).
O. Fotiy, A. Gumenchuk, I. Krasikova, and M. Popov, ‘‘On sums of narrow and compact operators,’’ Positivity 24, 69–80 (2020).
E. V. Kolesnikov, ‘‘Several order projections generated by ideals of a vector lattice,’’ Sib. Math. J. 36, 1342–1349 (1995).
E. V. Kolesnikov, ‘‘In the shadow of a positive operator,’’ Sib. Math. J. 37, 513–518 (1996).
M. A. Krasnosel’skij, P. P. Zabrejko, E. I. Pustil’nikov, and P. E. Sobolevskij, Integral Operators in Spaces of Summable Functions (Noordhoff, Leiden, 1976).
V. Mykhaylyuk, M. Pliev, and M. Popov, ‘‘The lateral order on Riesz spaces and orthogonally additive operators,’’ Positivity (2020). https://doi.org/10.1007/s11117-020-00761-x
M. Pliev and X. Fang, ‘‘Narrow orthogonally additive operators in lattice-normed spaces,’’ Sib. Math. J. 58, 134–141 (2017).
M. Pliev and M. Popov, ‘‘On extension of abstract Urysohn operators,’’ Sib. Math. J. 57, 552–557 (2016).
M. Pliev and K. Ramdane ‘‘Order unbounded orthogonally additive operators in vector lattices,’’ Mediter. J. Math. 15 (2) (2018).
M. Pliev and M. Weber, ‘‘Disjointness and order projections in the vector lattices of abstract Uryson operators,’’ Positivity 20, 695–707 (2016).
M. A. Pliev, F. Polat, and M. R. Weber, ‘‘Narrow and \(C\)-compact orthogonally additive operators in lattice-normed spaces,’’ Results Math. 74 (4) (2019).
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The author was supported by the Russian Foundation for Basic Research (grant no. 21-51-46006).
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(Submitted by A. B. Muravnik)
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Abasov, N.M. On a Band Generated by a Disjointness Preserving Orthogonally Additive Operator. Lobachevskii J Math 42, 851–856 (2021). https://doi.org/10.1134/S1995080221050024
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DOI: https://doi.org/10.1134/S1995080221050024