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 Sibirsk. Mat. Zh., 2017, Volume 58, Number 1, Pages 174–184 (Mi smj2850)

Narrow orthogonally additive operators in lattice-normed spaces

M. A. Plievab, X. Fangc

a Southern Mathematical Institute, Vladikavkaz, Russia
b Peoples' Friendship University of Russia, Moscow, Russia
c Department of Mathematics, Tongji University, Shanghai, China

Abstract: We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every $C$-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space $V$ into a Banach space $Y$. Furthermore, every dominated Urysohn operator from $V$ into a Banach sequence lattice $Y$ is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space $V$ into a Banach space with mixed norm $W$ implies the order narrowness of the least dominant of the operator.

Keywords: vector lattice, Banach lattice, lattice-normed space, orthogonally additive operator, dominated Urysohn operator, narrow operator.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-51-53119 The first author was supported by the Russian Foundation for Basic Research (Grant 15-51-53119) and by the Ministry of Education and Science of the Russian Federation (Agreement 02.a03.21.0008 of 24 June 2016). The second author was supported by the National Natural Science Foundation of China (Grant 11511130012).

DOI: https://doi.org/10.17377/smzh.2017.58.117

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English version:
Siberian Mathematical Journal, 2017, 58:1, 134–141

Bibliographic databases:

UDC: 517.98+519.46
MSC: 35R30

Citation: M. A. Pliev, X. Fang, “Narrow orthogonally additive operators in lattice-normed spaces”, Sibirsk. Mat. Zh., 58:1 (2017), 174–184; Siberian Math. J., 58:1 (2017), 134–141

Citation in format AMSBIB
\Bibitem{PliFan17} \by M.~A.~Pliev, X.~Fang \paper Narrow orthogonally additive operators in lattice-normed spaces \jour Sibirsk. Mat. Zh. \yr 2017 \vol 58 \issue 1 \pages 174--184 \mathnet{http://mi.mathnet.ru/smj2850} \crossref{https://doi.org/10.17377/smzh.2017.58.117} \elib{http://elibrary.ru/item.asp?id=29159913} \transl \jour Siberian Math. J. \yr 2017 \vol 58 \issue 1 \pages 134--141 \crossref{https://doi.org/10.1134/S0037446617010177} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000396065100017} \elib{http://elibrary.ru/item.asp?id=29482059} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85014643073} 

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This publication is cited in the following articles:
1. N. M. Abasov, M. A. Pliev, “O summe uzkogo i $C$-kompaktnogo operatorov”, Vladikavk. matem. zhurn., 20:1 (2018), 3–9
2. N. Abasov, M. Pliev, “On two definitions of a narrow operator on Köthe-Bochner spaces”, Arch. Math., 111:2 (2018), 167–176
3. M. Pliev, K. Ramdane, “Order unbounded orthogonally additive operators in vector lattices”, Mediterr. J. Math., 15:2 (2018), 55
4. M. A. Pliev, M. R. Weber, “Finite elements in some vector lattices of nonlinear operators”, Positivity, 22:1 (2018), 245–260
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