A Krengel Type Theorem for Compact Operators Between Locally Solid Vector Lattices

Abstract

Suppose $X$ and $Y$ are locally solid vector lattices. A linear operator $T:X\to Y$ is said to be $nb$-compact provided that there exists a zero neighborhood $U\subseteq X$, such that $\overline{T(U)}$ is compact in $Y$; $T$ is $bb$-compact if for each bounded set $B\subseteq X$, $\overline{T(B)}$ is compact. These notions are far from being equivalent, in general. In this paper, we introduce the notion of a locally solid $AM$-space as an extension for $AM$-spaces in Banach lattices. With the aid of this concept, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices. This extends [1, Theorem 5.7]} (established for compact operators between Banach lattices) to different classes of compact operators between locally solid vector lattices.