On limited and almost limited operators between Banach lattice
DOI:
https://doi.org/10.31489/2026m3/33-48Keywords:
Banach lattice, Banach space, Schur property, limited set, L-weakly compact set, semi-compact operator, Dunford–Pettis operator, collectively qualified set of operators, enveloping normAbstract
This article investigates theoretical questions concerning the class of (almost) limited operators on Banach lattices, focusing on their precise relations to L-weakly compact, semi-compact, and Dunford–Pettis operators. The authors establish several conditions under which bounded operators are AM-compact and limited, and characterize when regular operators are almost limited. Furthermore, several new notions of collectively qualified sets of operators are introduced. Using results obtained for individual operators, the authors derive new structural results concerning these collectively qualified sets. The lattice approximation property for collectively M-weakly compact families of operators is also thoroughly investigated. Finally, the paper examines some related topics in functional analysis, focusing on the specific behavior of various kinds of operators acting between Banach lattices and Banach spaces. The presented theorems extend existing results, providing deeper insights into the topological and order properties of bounded operators, and establishing new connections between collective compactness and lattice structures.
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