Geodesic
A Krengel type theorem for compact operators between locally solid vector lattices
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 76-80 
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/
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     title = {A {Krengel} type theorem for compact operators between locally solid vector lattices},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Aliprantis, C. D. and Burkinshaw, O., Positive Operators, Springer, 2006 | Zbl

[2] Zabeti, O., “$AM$-Spaces from a Locally Solid Vector Lattice Point of View with Applications”, Bulletin of the Iranian Mathematical Society, 47 (2021), 1559–1569 | DOI | MR | Zbl

[3] Aliprantis, C. D. and Burkinshaw, O., Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, 105, American Mathematical Society, Providence, 2003 | DOI | MR | Zbl

[4] Jameson, G. J. O., “Topological $M$-Spaces”, Mathematische Zeitschrift, 103 (1968), 139–150 | DOI | MR | Zbl

[5] Zabeti, O., “The Banach-Saks Property from a Locally Solid Vector Lattice Point of View”, Positivity, 25 (2021), 1579–1583 | DOI | MR | Zbl

[6] Jameson, G. J. O., Ordered Linear Spaces, Lecture Notes in Mathematics, 141, Springer, 1970 | DOI | MR | Zbl

[7] Erkursun-Ozcan, N., Anil Gezer, N. and Zabeti, O., “Spaces of $u\tau$-Dunford-Pettis and $u\tau$-Compact Operators on Locally Solid Vector Lattices”, Matematicki Vesnik, 71:4 (2019), 351–358 | MR | Zbl

[8] Troitsky, V. G., “Spectral Radii of Bounded Operators on Topological Vector Spaces”, PanAmerican Mathematical Journal, 11:3 (2001), 1–35 | MR | Zbl