弱紧“矩阵”、Fubini性质与稠密算子半群的扩张
Weakly Compact "Matrices", Fubini-Like Property and Extension of Densely Defined Semigroups of Operators
摘要：
把矩阵作为两个（一般是无限的）集合的笛卡儿积上的数值函数的同义词，一个简单的纯代数的“互反性质”说，行的集合跨越有限维空间当且仅当列的集合跨越有限维空间。类似的拓扑互易性质用于定义强紧和弱紧矩阵，其特征在于关于殆周期函数和关于紧算子的众所周知的基本事实。一些性质，特别是弱紧的情况下，进行了研究，如与矩阵的一般手段具有Fubini的性质。应用这些结果证明了在具有弱连续性的巴拿赫空间中有界稠定算子半群扩张到整半群的可能性。
英文摘要：
Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic "reciprocity property" says that the set of rows spans a finite-dim space iff the set of columns does so. Similar topological reciprocity properties serve to define strongly compact and weakly compact matrices, featured in the well-known basic facts about almost periodic functions and about compact operators. Some properties, especially for the weak compact case, are investigated, such as the connection with the matrix having a Fubini-like property for general means. These are applied to prove possibility of extension to the entire semigroup of bounded densely defined semigroups of operators in a Banach space with weak continuity properties.
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