This paper presents the first part of a study of functional models of selfadjoint and nonselfadjoint extensions $$\widetilde{A}$$ of symmetric and nonsymmetric operators A in a Hilbert space $$\mathfrak {H}$$ . The extensions will be considered in the framework of linear relations (which may also be interpreted as the graphs of multivalued operators) that are required to have a nonempty set of regular points $$\rho (\widetilde{A})$$ . In these models $$\mathfrak {H}$$ is modelled by a reproducing kernel Hilbert space $$\mathcal {H}$$ of vector valued holomorphic functions that are defined on some nonempty open set $$\Omega \subseteq \rho (\widetilde{A})$$ and $$\mathcal {H}$$ is invariant under the action of the (generalized) backward shift operator $$R_\alpha $$ for every $$\alpha \in \Omega $$ ; A is modelled by the operator $$\mathfrak {A}$$ of multiplication by the independent variable [i.e., $$(\mathfrak {A}f)(\lambda )=\lambda f(\lambda )$$ for $$f\in \mathcal {H}$$ for which $$\mathfrak {A}f\in \mathcal {H}$$ ]; and $$\widetilde{A}$$ is modelled by a linear relation $$\widetilde{\mathfrak {A}}$$ with the property that $$(\widetilde{\mathfrak {A}}-\alpha I)^{-1}=R_\alpha $$ for all points $$\alpha \in \Omega $$ .

中文翻译：

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希尔伯特空间中算子的函数模型及其多值扩展

本文介绍了 Hilbert 空间 $$\mathfrak {H}$$ 中对称和非对称算子 A 的 selfadjoint 和 nonselfadjoint 扩展 $$\widetilde{A}$$ 的函数模型研究的第一部分。扩展将在线性关系（也可以解释为多值运算符的图）的框架中考虑，该框架需要具有非空的正则点集 $$\rho (\widetilde{A})$$ 。在这些模型中，$$\mathfrak {H}$$ 由在一些非空开集 $$\Omega \subseteq \rho 上定义的向量值全纯函数的再现核希尔伯特空间 $$\mathcal {H}$$ 建模(\widetilde{A})$$ 和 $$\mathcal {H}$$ 在（广义）向后移位运算符 $$R_\alpha $$ 的作用下对于每个 $$\alpha \in \Omega $ 是不变的$; A 由运算符 $$\mathfrak {A}$$ 乘以自变量 [即，$$(\mathfrak {A}f)(\lambda )=\lambda f(\lambda )$$ 建模为 $ $f\in \mathcal {H}$$ 其中 $$\mathfrak {A}f\in \mathcal {H}$$ ]; 并且 $$\widetilde{A}$$ 由线性关系 $$\widetilde{\mathfrak {A}}$$ 建模，其性质为 $$(\widetilde{\mathfrak {A}}-\alpha I) ^{-1}=R_\alpha $$ 对于所有点 $$\alpha \in \Omega $$ 。