On operators dominated by Kantorovich-Banach operators and Levy operators in locally solid lattices

A linear operator T acting in a locally solid vector lattice (E,τ) is said to be: a Lebesgue operator, if Txα→τ0 for every net in E satisfying xα↓0; a KB-operator, if, for every τ-bounded increasing net xα in E+, there exists an x∈E with Txα→τTx; a quasi KB-operator, if T takes τ-bounded increasing nets in E+ to τ-Cauchy ones; a Levi operator, if, for every τ-bounded increasing net xα in E+, there exists an x∈E such that Txα→oTx; a quasi Levi operator, if T takes τ-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 L\'evi 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.