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Weak compactness of almost limited operators. (English) Zbl 1385.47011

Suppose that \(E\) is a Banach lattice and \(F\) is a \(\sigma\)-complete Banach lattice.A (continuous linear) operator \(T:E\rightarrow F\) is called almost limited if \(\left\| T^{\ast}(f_{n})\right\| \rightarrow0\) whenever \((f_{n})_{n}\) is a disjoint weak\(^{\ast}\) null sequence in \(F^{\ast}\). Theorem 5 asserts the equivalence of the following three conditions: (a) every almost limited operator \(T:E\rightarrow F\) is weakly compact; (b) every positive almost limited operator \(T:E\rightarrow F\) is weakly compact; (c) \(E\) is reflexive or the norm of \(F\) is order continuous. Theorem 9 offers the following characterization of Banach lattices with order continuous norm: If \(E\) is a \(\sigma\)-complete Banach lattice, then the norm of \(E\) is order continuous if and only if the square of every positive almost limited operator \(T:E\rightarrow E\) is weakly compact.

MSC:

47B07 Linear operators defined by compactness properties
46B42 Banach lattices
47B60 Linear operators on ordered spaces
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References:

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