Sums of lattice homomorphisms
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- by S. J. Bernau, C. B. Huijsmans and B. de Pagter PDF
- Proc. Amer. Math. Soc. 115 (1992), 151-156 Request permission
Abstract:
Let $E$ and $F$ be Riesz spaces and ${T_1},{T_2}, \ldots ,{T_n}$ be linear lattice homomorphisms (henceforth called lattice homomorphisms) from $E$ to $F$. If $T = \sum \nolimits _{i = 1}^n {{T_i}}$, then it is easy to check that $T$ is positive and that if ${x_0},{x_1}, \ldots {x_n} \in E$ and ${x_i} \wedge {x_j} = 0$ for all $i \ne j$, then $\wedge _{i = 0}^nT{x_i} = 0$. The purpose of this note is to show that if $F$ is Dedekind complete, the above necessary condition for $T$ to be be the sum of $n$ lattice homomorphisms is also sufficient. The result extends to sums of disjointness preserving operators, thereby leading to a characterization of the ideal of order bounded operators generated by the lattice homomorphisms.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 151-156
- MSC: Primary 46A40; Secondary 06F20, 47B60, 47B65
- DOI: https://doi.org/10.1090/S0002-9939-1992-1086322-8
- MathSciNet review: 1086322