Dunford-Pettis operators on Banach lattices
Dunford-Pettis operators on Banach lattices
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DOI：
10.1090/s0002-9947-1982-0670929-1
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发表时间：
1982
影响因子：
1.3
通讯作者：
C. Aliprantis;O. Burkinshaw
中科院分区：
文献类型：
--
作者：
C. Aliprantis;O. Burkinshaw
。考虑一个 Banach 格子 E 和两个满足 0 *c S =c T 的正算子 S,T: E -> E。在[2,3]中，我们研究了 T 是紧致（或弱紧致）算子的情况，并研究了这对由 T 主导的算子（例如 S）有什么影响。在本文中，我们扩展了这些技术并研究了有关 Dunford-Pettis 算子的类似问题。特别是，将对算子 T 给出条件，以确保 S（或 S 的某个幂）是 Dunford-Pettis 算子。作为示例，以下是处理这些问题的主要结果之一。定理。设 E 为 Banach 格，设 S,T: E -» E 为两个正算子，使得 0 < S =c T。如果 T 是紧算子，则 (1) S3 是紧算子（尽管 S2 不必是紧算子）； (2) S2 是 Dunford-Pettis 弱紧算子（尽管 S 不一定是）； (3) S 是弱 Dunford-Pettis 算子。在另一个方向上，我们的技术和结果将与 Dunford-Pettis 算子的晶格结构相关。例如，将证明在某些条件下，邓福德-佩蒂斯算子形成一个带。
. Consider a Banach lattice E and two positive operators S,T: E -> E that satisfy 0 *c S =c T. In [2,3] we examined the case when T is a compact (or weakly compact) operator and studied what effect this had on an operator (such as S) dominated by T. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator T, to ensure that S (or some power of S) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters. Theorem. Let E be a Banach lattice, and let S,T: E -» E be two positive operators such that 0 < S =c T. If T is compact then (1) S3 is a compact operator (although S2 need not be compact); (2) S2 is a Dunford-Pettis and weakly compact operator ( although S need not be); (3) S is a weak Dunford-Pettis operator. In another direction, our techniques and results will be related to the lattice stmcture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.