Unbounded Banach--Saks operators and unbounded Grothendieck operators on Banach lattices
Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 137-143
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Recently, there has been much attention to the ordered structures beyond Banach lattices. Moreover, we have many nice properties in Banach spaces that can be transformed naturally into Banach lattice cases. Therefore, combining these notions with order structure can produce nicer notions, as well. Suppose $E$ is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach–Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property. We characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding operator versions related to the Banach–Saks properties. Moreover, we consider different notions related to the Grothendieck property from an ordered attitude; then, we investigate operator versions of these concepts, as well. In particular, beside other results, we characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding bounded linear operators defined on the corresponding Banach lattices, as well.
@article{VMJ_2024_26_4_a11,
author = {R. Sabbagh and O. Zabeti},
title = {Unbounded {Banach--Saks} operators and unbounded {Grothendieck} operators on {Banach} lattices},
journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
pages = {137--143},
publisher = {mathdoc},
volume = {26},
number = {4},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/VMJ_2024_26_4_a11/}
}
TY - JOUR AU - R. Sabbagh AU - O. Zabeti TI - Unbounded Banach--Saks operators and unbounded Grothendieck operators on Banach lattices JO - Vladikavkazskij matematičeskij žurnal PY - 2024 SP - 137 EP - 143 VL - 26 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VMJ_2024_26_4_a11/ LA - en ID - VMJ_2024_26_4_a11 ER -
%0 Journal Article %A R. Sabbagh %A O. Zabeti %T Unbounded Banach--Saks operators and unbounded Grothendieck operators on Banach lattices %J Vladikavkazskij matematičeskij žurnal %D 2024 %P 137-143 %V 26 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/VMJ_2024_26_4_a11/ %G en %F VMJ_2024_26_4_a11
R. Sabbagh; O. Zabeti. Unbounded Banach--Saks operators and unbounded Grothendieck operators on Banach lattices. Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 137-143. http://geodesic.mathdoc.fr/item/VMJ_2024_26_4_a11/