On two definitions of a narrow operator on Köthe–Bochner spaces
On two definitions of a narrow operator on Köthe–Bochner spaces
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关于 Köthe-Bochner 空间上窄算子的两个定义
DOI：
10.1007/s00013-018-1172-2
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发表时间：
2018
影响因子：
0.6
通讯作者：
M. Pliev
中科院分区：
文献类型：
--
作者：
N. Abasov;M. Pliev
本文证明了：对于有限无原子测度空间$$（\Omega，\Sigma，\mu）$$（Ω，Σ，μ）上具有序连续范数的Köthe-Banach空间E和Banach空间X，Y，从Köthe-Bochner空间E（X）到Banach空间Y的窄算子类和弱泛函窄算子类是重合的.在一般情况下，在不假设E的范数序连续的情况下，从Köthe-Bochner空间E（X）到Banach空间Y的窄算子和弱泛函窄算子的定义等价当且仅当E（X）中的所有单元素的集合是稠密的.
We prove that for a Köthe–Banach space E with an order continuous norm over a finite atomless measure space $$(\Omega ,\Sigma ,\mu )$$(Ω,Σ,μ) and for Banach spaces X, Y, the classes of narrow and weakly functionally narrow operators from a Köthe–Bochner space E(X) to a Banach space Y are coincident. We also obtain that in the general case, without the assumption of order continuity of the norm of E, the definitions of narrow and weakly functionally narrow operators from a Köthe–Bochner space E(X) to a Banach space Y are equivalent if and only if the set of all simple elements is dense in E(X).