On operators dominated by Kantorovich–Banach operators and Lévy operators in locally solid lattices

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 Lévi 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 Lévi operators in locally solid vector lattices. Moreover, some properties of Lebesgue operators, Lévi 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.