广义Nevanlinna函数在无穷远处解析的逆函数
Inverse of generalized Nevanlinna function that is holomorphic at infinity
AI总结：
本文研究了在无穷远处解析的广义Nevanlinna函数的逆函数，通过新的算子表示方法，将逆函数分解为两个简单组成部分，从而研究其性质。
摘要：
设$(\mathcal{H},(.,.))$为希尔伯特空间，令$\mathcal{L}(\mathcal{H})$为$\mathcal{H}$中的有界算子线性空间。本文研究属于广义Nevanlinna类$\mathcal{N}_κ (\mathcal{H})$的$\mathcal{L}(\mathcal{H})$-值函数$Q$，其中$κ$为非负整数。该类函数在复平面$\mathbb{C}\backslash\mathbb{R}$上是亚纯的，满足$Q(z)^{*}=Q(\bar{z})$，其核$\mathcal{N}_{Q}(z,w):=\frac{Q(z)-{Q(w)}^{*}}{z-\bar{w}}$具有$κ$个负平方。重点研究在无穷远处解析的函数$Q \in \mathcal{N}_κ (\mathcal{H})$。在导数在无穷远处$Q^{'}(\infty):=\lim\limits_{z\to \infty}zQ(z)$为有界可逆算子的条件下，获得逆函数$\hat{Q}(z):=-{Q(z)}^{-1}$的新算子表示。结果表明，$\hat{Q}$可表示为$\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2}$，其中$\hat{Q}_{i}\in \mathcal{N}_{κ_{i}}(\mathcal{H})$，且$κ_{1}+κ_{2}=κ$。这种分解使我们能够通过研究简单的组成部分$\hat{Q}_{1}$和$\hat{Q}_{2}$来研究$Q$和$\hat{Q}$的性质。
英文摘要：
Let $\left(\mathcal{H},\left(.,.\right)\right)$ be a Hilbert space and let $\mathcal{L}\left(\mathcal{H}\right)$ be the linear space of bounded operators in $\mathcal{H}$. In this paper, we deal with $\mathcal{L}(\mathcal{H})$-valued function $Q$ that belongs to the generalized Nevanlinna class $\mathcal{N}_κ (\mathcal{H})$, where $κ$ is a non-negative integer. It is the class of functions meromorphic on $C \backslash R$, such that $Q(z)^{*}=Q(\bar{z})$ and the kernel $\mathcal{N}_{Q}\left( z,w \right):=\frac{Q\left( z \right)-{Q\left( w \right)}^{\ast }}{z-\bar{w}}$ has $κ$ negative squares. A focus is on the functions $Q \in \mathcal{N}_κ (\mathcal{H})$ which are holomorphic at $ \infty$. A new operator representation of the inverse function $\hat{Q}\left( z \right):=-{Q\left( z \right)}^{-1}$ is obtained under the condition that the derivative at infinity $Q^{'}\left( \infty\right):=\lim\limits_{z\to \infty}{zQ(z)}$ is boundedly invertible operator. It turns out that $\hat{Q}$ is the sum $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in \mathcal{N}_{κ_{i}}\left( \mathcal{H} \right)$ that satisfies $κ_{1}+κ_{2}=κ$. That decomposition enables us to study properties of both functions, $Q$ and $\hat{Q}$, by studying the simple components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
