“矩性质蕴含次正规性”及单回路图上的复合算子单射性问题的解

中文摘要 AI 辅助

我们考虑 $L^2(\mu)$ 中的复合算子，其中 $\mu$ 是一个离散测度。我们的主要结果表明，对于每个整数 $p\ge2$，存在一个非次正规的复合算子，它生成 Stieltjes 矩序列，且其符号诱导一个有向图，该图由一个长度为 $p$ 的回路和一条无限分支组成。相反，如果符号诱导的有向图由一个环和一条无限分支组成，则 Stieltjes 矩序列的生成确实蕴含次正规性。这两个结果完全解决了 [Adv. Math. 310 (2017), 484-556] 中针对具有任意长度回路的单分支有向图整个族留下的“矩性质蕴含次正规性”问题。将它们与该文中的例子结合表明，对于由一个环和有限多条无限分支组成的有向图，$2$ 是矩性质不蕴含次正规性的最小分支数。最后，我们构造了一个非单射的复合算子，它生成 Stieltjes 矩序列，且其符号诱导一个有向图，该图由一个环、一条无限分支和一片叶子组成。该算子对上述论文中提出的单射性问题给出了否定回答。两种构造都使用了任意归一化的 S-不确定 Stieltjes 矩序列的 Krein 和 Friedrichs 测度。

英文摘要

We consider composition operators in $L^2(μ)$, where $μ$ is a discrete measure. Our main result shows that, for every integer $p\ge2$, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length $p$ and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the ``moment property implies subnormality'' problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.

发表机构

• Uniwersytet Rolniczy w Krakowie（克拉科夫农业大学）
• Uniwersytet Jagielloński（雅盖隆大学）
• Kyungpook National University（庆北国立大学）

机构由 AI 辅助整理，请以论文原文为准。