A Krengel type theorem for compact operators between locally solid vector lattices

Suppose X and Y are locally solid vector lattices. A linear operator T:X→Y is said to be nb-compact provided that there exists a zero neighborhood U⊆X, such that T(U) is compact in Y; T is bb-compact if for each bounded set B⊆X, 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.