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Proceedings of the American Mathematical Society

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On weakly compact operators on Banach lattices

Authors: C. D. Aliprantis and O. Burkinshaw
Journal: Proc. Amer. Math. Soc. 83 (1981), 573-578
MSC: Primary 47B55; Secondary 46B30
MathSciNet review: 627695
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Abstract: Consider a Banach lattice $ E$ and an order bounded weakly compact operator $ T:E \to E$. The purpose of this note is to study the weak compactness of operators that are related with $ T$ in some order sense. The main results are the following.

(1) If $ T$ is a positive weakly compact operator and an operator $ S:E \to E$ satisfies $ 0 \leqslant S \leqslant T$, then $ {S^2}$ is weakly compact. (Examples show that $ S$ need not be weakly compact.)

(2) If $ T$ and $ S$ are as in (1) and either $ S$ is an orthomorphism or $ E$ has an order continuous norm, then $ S$ is weakly compact.

(3) If $ E$ is an abstract $ L$-space and $ T$ is weakly compact, then the modulus $ \vert T\vert$ is weakly compact.

References [Enhancements On Off] (What's this?)

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Keywords: Banach lattice, positive operator, weakly compact operator
Article copyright: © Copyright 1981 American Mathematical Society

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