On Operators Dominated by the Kantorovich–Banach and Levi Operators in Locally Solid Lattices

Abstract

A linear operator $T$ in a locally solid vector lattice $(E,\tau)$ is a Lebesgue operator, if $Tx_{\alpha}\overset{\tau}{\rightarrow}0$ 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 $x\in E$ with $Tx_{\alpha}\overset{\tau}{\rightarrow}Tx$; a quasi $KB$-operator, if $T$ takes $\tau$-bounded increasing nets in $E_{+}$ to $\tau$-Cauchy nets; a Levi operator, if for every $\tau$-bounded increasing net $x_{\alpha}$ in $E_{+}$ there exists $x\in E$ such that $Tx_{\alpha}\overset{o}{\rightarrow}Tx$; and a quasi Levi operator, if $T$ takes $\tau$-bounded increasing nets in $E_{+}$ to $o$-Cauchy ones. We address the domination problem for the quasi $KB$-operators and quasi Levi operators in locally solid vector lattices. Moreover, under study are some properties of Lebesgue, Levi, and $KB$-operators. In particular, we prove that the vector space of regularly Lebesgue operators is a subalgebra of the algebra of all regular operators.