前向算子幺半群
Forward operator monoids
AI总结：
本文提出并研究前向算子幺半群，重点探讨其相关的循环向量、广义内向量及 aleph 向量，并证明 aleph 向量的唯一性及其在刻画向量中的应用，最后将其与 Riemann Hypothesis 等著名问题相联系。
摘要：
本文研究在复合运算下具有严格幺半群结构的 Hilbert 空间算子集合。这些集合可以被视为离散的含幺半群，其中除了仅包含单位元的平凡子集外，没有任何子集在复合运算下是封闭的。我们将这些集合称为前向算子幺半群。尽管传统上并未以这种视角来审视，这种类型的幺半群却出现在分析的诸多领域中；正如我们将要看到的，它们与几个著名问题密切相关。本工作的主要目的是研究与给定算子幺半群相关的三类向量：循环向量、广义内向量，以及既是循环又是内的向量（我们将最后一类称为 aleph 向量）。我们证明，当 aleph 向量存在时，它们在相差一个乘法常数的意义下是唯一的。我们还展示了，在最小二乘问题的视角下，如何利用 aleph 向量来刻画所有的循环向量和内向量。作为该理论的应用，我们考虑了与 Periodic Dilation Completeness Problem 和 Riemann Hypothesis 相关的幺半群。在恢复一些已知结果之后，我们基于 Báez-Duarte 和 Noor 的工作，给出了 Riemann Hypothesis 的进一步重新表述。
英文摘要：
This work studies collections of Hilbert space operators which possess a strict monoid structure under composition. These collections can be thought of as discrete unital semigroups for which no subset of the collection is closed under composition (apart from the trivial subset containing only the identity). We call these collections $\textit{forward operator monoids}$. Although not traditionally painted in this light, monoids of this flavor appear in many areas of analysis; as we shall see, they are intimately linked to several well-known problems. The main aim of this work is to study three classes of vectors associated to a given operator monoid: cyclic vectors, generalized inner vectors, and vectors which are both cyclic and inner (we refer to this last class as aleph vectors). We show, when they exist, aleph vectors are unique up to a multiplicative constant. We also show, under the lens of a least-squares problem, how an aleph vector can be used to characterize all cyclic and inner vectors. As an application of this theory, we consider monoids related to the Periodic Dilation Completeness Problem and the Riemann Hypothesis. After recovering some known results, we conclude by giving a further reformulation of the Riemann Hypothesis based on works of Báez-Duarte and Noor.
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