Converses for the Dodds–Fremlin and Kalton–Saab theorems
Converses for the Dodds–Fremlin and Kalton–Saab theorems
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DOI：
10.1017/s0305004100074752
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发表时间：
1996-07
影响因子：
0.8
通讯作者：
A. Wickstead
中科院分区：
文献类型：
--
作者：
A. Wickstead
标题中的两个定理给出了Banach格E和F上的条件，在该条件下，从E到F的正算子，被另一个具有某种性质的正算子支配，也必须具有该性质。Dodds-Fremlin定理指出，如果E′和F都有序连续范数，这对紧性是正确的，而Kalton-Saab定理建立了Dunford-Pettis算子的这样一个结果，如果F有序连续范数。这些结果最初在[3]和[5]中分别提供了它们的全部一般性，而非常可读的证明可以在[2]的第5章或[6]的§3·7中找到。
The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].