Abstract
Projections onto several special subsets in the Dedekind complete vector lattice of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F are considered and some new formulas are provided.
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Notes
\((C_{1})\) and \((C_{2})\) are called the Carathéodory conditions.
An element \(u\in F_+\) is a weak order unit if \(\{u\}^{\bot \bot }=F\), i.e. except 0 there are no elements in F which are disjoint to u.
A weak order unit we need for applying this theorem is \(u+v\), where u, as was already mentioned, is a weak order unit in \(\{Sx+Tx\}^{\bot \bot }\) and v is some weak order unit in \(\{Sx+Tx\}^\bot \).
For example, \(y=x\) and arbitrary \(\rho \in \mathfrak {P}(F)\).
In both cases without touching the term \(\varepsilon nS_0x\).
By using the relations \(\rho \circ \rho =\rho \) and \(\rho \circ \rho ^\bot =0\).
For \(S\in \mathcal {U}_{+}(E,F)\) the projections onto the bands \(\{S\}^{\bot \bot }\) and \(\{S\}^\bot \) are denoted by \(\pi _S\) and \(\pi _S^\bot \), respectively.
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M. A. Pliev was supported by the Russian Foundation of Basic Research, the Grant Number 14-01-91339.
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Pliev, M.A., Weber, M.R. Disjointness and order projections in the vector lattices of abstract Uryson operators. Positivity 20, 695–707 (2016). https://doi.org/10.1007/s11117-015-0381-1
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DOI: https://doi.org/10.1007/s11117-015-0381-1