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Atomic operators in vector lattices. (English) Zbl 1520.47073

The study of orthogonally additive operators (and their various subclasses) between vector lattices has garnered a lot of attention since their introduction in the 1990s [J. M. Mazón and S. Segura de León, Rev. Roum. Math. Pures Appl. 35, No. 4, 329–353 (1990; Zbl 0723.47047); Rev. Roum. Math. Pures Appl. 35, No. 5, 431–449 (1990; Zbl 0717.47031)]. Let \(E\) and \(F\) be vector lattices and \(\Phi\) be a Boolean homomorphism from Boolean algebra \(\mathfrak B(E)\) to the Boolean algebra \(\mathfrak B(F)\). N. Abasov and M. Pliev [J. Math. Anal. Appl. 455, No. 1, 516–527 (2017; Zbl 1459.47020)] introduced the class of \(\Phi\)-operators. An operator (not necessarily linear) \(T:E\to F\) is called a \(\Phi\)-operator if \[ T\circ \pi=\Phi(\pi)T\text{ for every order projection }\pi\in \mathfrak B(E). \] If \(E\) has the principal projection property, then the \(\Phi\)-operators from \(E\) to \(F\) form a subclass of the orthogonally additive operators.
The paper under review calls an operator \(T:E\to F\) atomic if there exists a Boolean homomorphism \(\Phi:\mathfrak B(E)\to \mathfrak B(F)\) such that \(T\) is a \(\Phi\)-operator. Properties of the set of all atomic operators from \(E\) to \(F\) subordinate to \(\Phi\) and of minimal extensions of atomic operators are studied. Furthermore, continuous atomic operators between spaces of real-valued measurable functions are characterized (Theorem 3.2).

MSC:

47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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References:

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