The subnormal completion problem on a directed tree is to determine, given a collection of weights on a subtree, whether the weights may be completed to the weights of a subnormal weighted shift on the directed tree. We study this problem on a directed tree with a single branching point, $$\eta $$ η branches and the trunk of length 1 and its subtree which is the “truncation” of the full tree to vertices of generation not exceeding 2. We provide necessary and sufficient conditions written in terms of two parameter sequences for the existence of a subnormal completion in which the resulting measures are 2-atomic. As a consequence, we obtain a solution of the subnormal completion problem for this pair of directed trees when $$\eta < \infty $$ η < ∞ . If $$\eta =2$$ η = 2 , we present a solution written explicitly in terms of initial data.

中文翻译：

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有向树上加权移位的次正规完成问题，II

有向树上的次正规完成问题是确定给定子树上的权重集合，权重是否可以完成到有向树上的次正规加权移位的权重。我们在具有单个分支点、$$\eta $$ η 个分支和长度为 1 的树干及其子树上研究这个问题，该子树是完整树到不超过 2 的生成顶点的“截断”。我们提供根据两个参数序列编写的必要和充分条件，用于存在次正规完成，其中产生的措施是 2 原子的。因此，当 $$\eta < \infty $$ η < ∞ 时，我们获得了这对有向树的次正规完成问题的解决方案。如果 $$\eta =2$$ η = 2 ，我们提出了一个根据初始数据明确编写的解决方案。