In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety $\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})$ in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators $B_{1},\ldots,B_{n}$, where the n-tuple $(B_{1},\ldots,B_{n})$ is the universal model associated with the abstract noncommutative variety $\mathcal{V}_{f,\varphi,\mathcal{I}}$.

中文翻译：

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约束特征函数，多变量插值和不变子空间

在本文中，我们给出了非交换变量$ \ mathcal {V} _ {f，\ varphi，\ mathcal {I}}（\ mathcal {H}）中算子的完全非对等对称n元组的函数模型定理$在受约束的特征函数方面。作为一个应用，我们证明了约束特征函数对于此类元素是一个完整的ary不变式，可以将其视为经典Sz.-Nagy-Foiaş泛函模型的完全非contract式收缩的非交换式。另一方面，我们提供了一个Sarason型换向提升定理。应用该结果，我们可以解决设置中的Nevanlinna–Pick型插值问题。此外，我们还获得了在运算符$ B_ {1}，\ ldots，B_ {n} $下的联合不变子空间的Beurling型特征，其中n元组$（B_ {1}，\ ldots，