环面 2 - 等距算子值双变量加权移位的柯西对偶次正规性
The Cauchy dual subnormality problem in two variables
AI总结：
研究解析左逆交换对环面 2 - 等距算子，满足联合核条件时与可逆权重算子值加权移位酉等价，初始权重交换时柯西对偶联合次正规，还构造反例说明初始权重不交换时柯西对偶非联合次正规。
摘要：
在本文中，我们证明如果\(\mathbf{T}=(T_1,T_2)\)是满足联合核条件的解析左逆交换对环面\(2 -\)等距算子，那么它酉等价于具有可逆权重\(\{W_{I}^{(j)}:j = 1,2\}_{I\in\mathbb{Z}_{+}^2}\)的算子值加权移位，其中初始权重\(W_{0,0}^{(1)}\)和\(W_{0,0}^{(2)}\)是正算子。此外，如果这些初始权重交换，那么柯西对偶\(\mathbf{T}':=(T_1',T_2')\)是联合次正规的。我们还构造了一个初始权重不交换且相应柯西对偶不是联合次正规的例子。
英文摘要：
The Cauchy dual subnormality problem asks whether the Cauchy dual of a $2$-isometry is subnormal. This problem has attracted considerable attention due to its connections with moment theory, operator theory, and complex function theory. In particular, it has been closely related to operator-theoretic models on function spaces such as de Branges--Rovnyak spaces and Dirichlet-type spaces. In this paper, we investigate the Cauchy dual subnormality problem in the multivariable setting of toral $2$-isometries. We show that if $\mathbf{T} = (T_1, T_2)$ is an analytic left-inverse commuting pair of toral $2$-isometries satisfying the joint kernel condition, then it is unitarily equivalent to an operator-valued weighted shift with invertible weights $\{W_{I}^{(j)}:j=1,2,I\in \mathbb{Z}_{+}^2\},$ where the initial weights $W_{0,0}^{(1)}$ and $W_{0,0}^{(2)}$ are positive operators. Moreover, if these initial weights commute, then the Cauchy dual $\mathbf{T}' := (T_1', T_2')$ is subnormal. We also construct an example in which the initial weights do not commute, and the corresponding Cauchy dual tuple $\mathbf{T'}=(T_1',T_2')$ fails to be subnormal. Finally, we establish a complete set of unitary invariants for pure toral $2$-isometries.
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