Atomic Operators in Vector Lattices
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Atomic Operators in Vector Lattices Ralph Chill
and Marat Pliev
Abstract. In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism Φ from the Boolean algebra B(E) of all order projections on E to B(F ) such that T π = Φ(π)T for every order projection π ∈ B(E). We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F . We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space. Mathematics Subject Classification. Primary 46B99; Secondary 47B38. Keywords. Orthogonally additive operator, atomic operator, disjointness preserving operator, nonlinear superposition operator, Boolean homomorphism, vector lattice, order ideal.
1. Introduction and Preliminaries Local operators and, more generally, atomic operators in classical function spaces find numerous applications in control theory, the theory of dynamical systems and the theory of partial differential equations (see [6,19,23]). The concept of a local operator was in the context of vector lattices first introduced in [22]. It is an abstract form of the well-known property of a nonlinear superposition operator and can be stated in the following form: the value of the image function on a certain set depends only on the values of the preimage function on the same set. In this article, we analyse the notion of an atomic operator in the framework of the theory of vector lattices and The authors are grateful for support by the Deutsche Forschungsgemeinschaft (Grant CH 1285/5-1, Order preserving operators in problems of optimal control and in the theory of partial differential equations) and by the Russian Foundation for Basic Research (Grant number 17-51-12064).
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orthogonally additive operators. Today, the theory of orthogonally additive operators in vector lattices is an active area in functional analysis; see for instance [1,2,7,8,10,11,13,14,16–18,25]. Abstract results of this theory can be applied to the theory of nonlinear integral operators [12,21], and there are connections with problems of convex geometry [24]. Let us introduce some basic facts concerning vector lattices and orthogonally additive operators. We assume that the reader is acquainted with the theory of vector lattices and Boolean algebras. For the standard information, we refer to [3,4,9]. All vector lattices below are assumed to be Archimedean. Let E be a vector lattice. A net (xα )α∈Λ in E order converges to an (o)
element x ∈ E (notation xα −→ x) if there exists a net
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