In this paper, we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space $\mathcal{H}(\Delta) \equiv H_E^2 \ominus \Delta H_{E}^2$, for some inner function $\Delta$. Our first question calls for a description of the set $F$ in $H_E^2$ such that $\mathcal{H}(\Delta)=E_F^*$, where $E_F^*$ denotes the smallest backward shift-invariant subspace containing the set $F$. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong $L^2$-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? As we know, the kernel of a Hankel operator is shift-invariant, so the above question is equivalent to seeking a solution to the equation $\ker H_{\Phi}^*=\Delta H_{E^{\prime}}^2$, where $\Delta$ is an inner function satisfying $\Delta^* \Delta=I_{E^{\prime}}$ almost everywhere on the unit circle $\mathbb{T}$ and $H_{\Phi}$ denotes the Hankel operator with symbol $\Phi$. Consideration of the above question on the structure of shift-invariant subspaces leads us to study and coin a new notion of "Beurling degree" for an inner function. We then establish a deep connection between the spectral multiplicity of the model operator and the Beurling degree of the corresponding characteristic function. At the same time, we consider the notion of meromorphic pseudo-continuations of bounded type for operator-valued functions, and then use this notion to study the spectral multiplicity of model operators (truncated backward shifts) between separable complex Hilbert spaces. In particular, we consider the multiplicity-free case.

中文翻译：

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无限重数的 Beurling-Lax-Halmos 定理

在本文中，我们考虑了从 Beurling-Lax-Halmos 定理中出现的几个问题，该定理表征了向量值哈代空间的移不变子空间。Beurling-Lax-Halmos 定理指出，对于某些内部函数，向后平移不变的子空间是模型空间 $\mathcal{H}(\Delta) \equiv H_E^2 \ominus \Delta H_{E}^2$ $\Delta$。我们的第一个问题要求描述 $H_E^2$ 中的集合 $F$，使得 $\mathcal{H}(\Delta)=E_F^*$，其中 $E_F^*$ 表示最小的后移不变式包含集合 $F$ 的子空间。在我们寻求这个问题的一般解决方案时，我们自然会考虑到算子值强 $L^2$ 函数的规范分解。接下来，我们问：每个移不变子空间都是（可能是无界的）Hankel 算子的核吗？据我们所知，Hankel算子的核是平移不变的，所以上面的问题等价于求方程$\ker H_{\Phi}^*=\Delta H_{E^{\prime}}^2$的解，其中 $\Delta$ 是满足 $\Delta^* \Delta=I_{E^{\prime}}$ 几乎所有单位圆 $\mathbb{T}$ 和 $H_{\Phi}$ 表示的内部函数带有符号 $\Phi$ 的 Hankel 算子。考虑上述关于移不变子空间结构的问题，我们研究并创造了一个新的内函数“Beurling 度”概念。然后，我们在模型算子的谱多重性和相应特征函数的 Beurling 度之间建立了深层联系。同时，我们考虑运算符值函数的有界类型的亚纯伪延续的概念，然后使用这个概念来研究可分离复 Hilbert 空间之间模型算子的谱多样性（截断后向位移）。特别地，我们考虑无多重性的情况。