与正规算子平方根相关算子的模型理论
Model Theory for Operators Related to Square Roots of Normal Operators
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中文总结 AI 辅助
研究与正规算子平方根相关算子，证明循环正规算子平方根及相关循环算子与特定向量值空间上块乘法算子酉等价，考虑向量值哈代空间界点赋值，得出相关算子有非平凡不变子空间及最小n-正规扩张唯一性等结论。
中文摘要 AI 辅助
本文证明了循环正规算子的平方根酉等价于具有2-正规符号的向量值勒贝格空间上的块乘法算子。此外，表明允许循环2-正规扩张的循环算子酉等价于具有2-正规符号的相应向量值哈代空间上的块乘法算子。考虑向量值哈代空间中界点赋值的存在性，作为应用，证明允许循环2-正规扩张的算子有非平凡不变子空间。还研究了具有循环2-正规扩张的算子的最小n-正规扩张的唯一性。
英文摘要
In this paper we prove that a square root of a cyclic normal operator is unitarily equivalent to a block multiplication operator on a vector-valued Lebesgue space with a $2$--normal symbol. In addition, we show that a cyclic operator which admits a cyclic $2$--normal extension is unitarily equivalent to a block multiplication operator on a corresponding vector-valued Hardy space with $2$--normal symbol. We consider the existence of bounded point evaluations in the vector-valued Hardy space setting; as an application, we prove that an operator admitting a cyclic $2$--normal extension has nontrivial invariant subspaces. We also study uniqueness for minimal $n$--normal extensions of operators that have cyclic $2$--normal extensions.
