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A note on disjointness preserving operators. (English) Zbl 0541.47032

This paper is concerned with the automatic order boundedness of disjointness preserving operators on Riesz spaces (vector lattices). A linear operator T from Archimedean Riesz space L into Archimedean Riesz space M is called disjointness preserving if \(\inf(| f|,| g|)=0\) in L implies that \(\inf(| Tf|,| Tg|)=0\) in M. The basic result of the paper is that, given such a disjointness preserving operator T and given 0\(\leq u\leq e\) in L, there exist sequences \(\{f_ n\}\) and \(\{g_ n\}\) in L such that \(f_ n,g_ n\to 0\) (e- uniformly) and \((| Tu| -| Te|)^+\leq | Tf_ n| +| Tg_ n|\) for all n. First of all, this theorem provides a simple proof of a result of Yu. A. Abramovich [Indag. Math. 45, 265-279 (1983; Zbl 0527.47025)] to the effect that a disjointness preserving operator with the property that \(\inf_ n(| Tf_ n| +| Tg_ n|)=0\) in M whenever \(f_ n,g_ n\to 0\) (rel. uniformly) in L, is order bounded. Another application of the theorem shows that any disjointness preserving operator from a Banach lattice L into a normed Riesz space M is order bounded on some order dense ideal in L. This last result implies in particular that a band preserving operator (i.e., a linear operator T such that \(\inf(| Tf|,| g|)=0\) whenever \(\inf(| f|,| g|)=0)\) in a Banach lattice is order bounded, and hence norm bounded, which is a result of Yu. A. Abramovich, A. I. Veksler and A. V. Koldunov [Dokl. Nauk SSSR 248, 1033-1036 (1979; Zbl 0445.46017)].

MSC:

47B60 Linear operators on ordered spaces
47A40 Scattering theory of linear operators
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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