REPRESENTATION OF ATOMIC OPERATORS AND EXTENSION PROBLEMS
REPRESENTATION OF ATOMIC OPERATORS AND EXTENSION PROBLEMS
复制标题
原子运算符的表示和扩展问题
DOI：
10.1017/s001309150200072x
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发表时间：
2004
影响因子：
0.7
通讯作者：
E. Stepanov
中科院分区：
文献类型：
--
作者：
E. Stepanov
摘要Drakhlin、Ponosov和Stepanov在2002年的一篇论文中引入了可测函数空间之间的原子算子的概念，以提供对应用有用的局部算子的合理推广。已经表明，粗略地说，原子算子相当于具有移位的局部算子的合成。一个自然的问题是，当一个连续测度原子算子可以表示为一个由Carathéodory函数生成的Nemytskii复合算子和一个移位算子的复合时。在本文中，我们将表明，这个问题的答案是内在的可能性，延长原子算子的连续性，从一个空间的功能可测相对于一些$\sigma$-代数到一个更大的空间的功能可测相对于一个更大的$\sigma$-代数，以及扩展任何$\sigma$-同态从一个较小的措施代数的$\sigma$-同态上的一个较大的措施代数的可能性。我们精确地刻画了相应的$\sigma$-代数上的条件，它提供了这样的可能性，并诱导了上述表示问题的肯定答案。AMS 2000数学科目分类：小学47 B38; 47 A67; 34 K 05
Abstract The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem. AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05