A Krengel type theorem for compact operators between locally solid vector lattices
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 76-80
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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.
@article{VMJ_2023_25_3_a5,
author = {O. Zabeti},
title = {A {Krengel} type theorem for compact operators between locally solid vector lattices},
journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
pages = {76--80},
publisher = {mathdoc},
volume = {25},
number = {3},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a5/}
}
O. Zabeti. A Krengel type theorem for compact operators between locally solid vector lattices. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 76-80. http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a5/