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DOI: 10.46698/f552500053031h On Operators Dominated by KantorovichBanach Operators and Levy Operators in Locally Solid Lattices
Abstract:
A linear operator \(T\) acting in a locally solid vector lattice \((E,\tau)\) is said to be: a Lebesgue operator, if \(Tx_\alpha\stackrel{\tau}{\to}0\) for every net in \(E\) satisfying \(x_\alpha\downarrow 0\); a \(KB\)operator, if, for every \(\tau\)bounded increasing net \(x_\alpha\) in \(E_+\), there exists an \(x\in E\) with \(Tx_\alpha\stackrel{\tau}{\to}Tx\); a quasi \(KB\)operator, if \(T\) takes \(\tau\)bounded increasing nets in \(E_+\) to \(\tau\)Cauchy ones; a Levi operator, if, for every \(\tau\)bounded increasing net \(x_\alpha\) in \(E_+\), there exists an \(x\in E\) such that \(Tx_\alpha\stackrel{o}{\to}Tx\); a quasi Levi operator, if \(T\) takes \(\tau\)bounded increasing nets in \(E_+\) to \(o\)Cauchy ones. The present article is devoted to the domination problem for the quasi \(KB\)operators and the quasi Levi operators in locally solid vector lattices. Moreover, some properties of Lebesgue operators, Levi operators, and \(KB\)operators are investigated. In particularly, it is proved that the vector space Lebesgue operators is a subalgebra of the algebra of all regular operators.
Keywords: locally solid lattice, Lebesgue operator, Levi operator, \(KB\)operator, lattice homomorphism
Language: Russian
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For citation: Gorokhova, S. G. and Emelyanov, E. Y. On Operators Dominated by KantorovichBanach Operators and Levy Operators in Locally Solid Lattices, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 5562 (in
Russian). DOI 10.46698/f552500053031h ← Contents of issue 
 

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