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Taurida Journal of Computer Science Theory and Mathematics, 2018, Issue 3, Pages 104–111
(Mi tvim55)
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On laterally continuous orthogonally additive operators
M. A. Pliev Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz
Abstract:
The aim of this article is to consider some problems of the theory
of orthogonally additive operators in vector lattices. Order bounded
orthogonally additive operators acting between vector lattices were
introduced and studied in 1990 by Mazón and Segura de León.
Recently, a new class of orthogonally additive operators in vector
lattices where the condition of order boundness of an operator is
replaced with a much weaker property was investigated by the author
of these notes and Ramdane. It is worth to note that today the
theory of orthogonally additive operators is an area of the intense
study. Let $E$ be a vector lattice and $F$ a real linear space. An
operator $T:E\rightarrow F$ is said to be orthogonally additive if
$T(x+y)=T(x)+T(y)$ whenever $x,y\in E$ are disjoint. Evidently from
the definition that $T(0)=0$. It is clear that the set of all
orthogonally additive operators is a real vector space with respect
to the natural linear operations. Let $E$ and $F$ be vector
lattices. We say that an orthogonally additive operator
$T:E\rightarrow F$ is positive if $T(E)\subset F_{+}$ and we say
that an orthogonally additive operator $T:E\rightarrow F$ is regular
if $T=S_{1}-S_{2}$ for some positive orthogonally additive operators
$S_{i}:E\to F$, $i\in{1,2}$. In this paper we investigate the band
of laterally continuous operators in the vector lattice of all
regular orthogonally additive operators between vector lattices $E$
and $F$. We show that the band which is disjoint to the band
generated by all singular orthogonally additive operators coincides
with the band of all laterally continuous orthogonally additive
operators.
Keywords:
Orthogonally additive operator, regular operator,
laterally continuous operator, singular operator, orthogonally
additive map, vector lattice, lateral ideal.
Citation:
M. A. Pliev, “On laterally continuous orthogonally additive operators”, Taurida Journal of Computer Science Theory and Mathematics, 2018, no. 3, 104–111
Linking options:
https://www.mathnet.ru/eng/tvim55 https://www.mathnet.ru/eng/tvim/y2018/i3/p104
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