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ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online) | |
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DOI: 10.46698/f5525-0005-3031-h On Operators Dominated by Kantorovich-Banach 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
For citation: Gorokhova, S. G. and Emelyanov, E. Y. On Operators Dominated by Kantorovich-Banach Operators and Levy Operators in Locally Solid Lattices, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 55-62 (in
Russian). DOI 10.46698/f5525-0005-3031-h
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