论文标题
无限大态的广义nevanlinna函数的倒数
Inverse of generalized Nevanlinna function that is holomorphic at infinity
论文作者
论文摘要
令$ \ left（\ Mathcal {h}，\ left（。，。\ right）\ right）$为Hilbert Space，让$ \ Mathcal {l} \ left（\ Mathcal {h} \ right）$是$ \ \ nathcal {h h} $的线性操作员的线性空间。在本文中，我们处理属于广义nevanlinna class $ \ nevanlinna class $ \ mathcal {n}_κ（\ mathcal {h}）$的$ \ MATHCAL {l}（\ MATHCAL {h}）$ - 值函数$ Q $，其中$κ$是一个非nenegation integer。它是$ c \ backslash r $上的函数类杂种类，因此$ q（z）^{*} = q（\ bar {z}）$和kernel $ \ nathcal {n} _ {q} _ {q} _ {q} \ left（z，w \ w \ right） \ right）}}^{\ ast}}} {z- \ bar {w}} $具有$κ$ panist Squares。重点是在\ Mathcal {n}_κ（\ Mathcal {h}）$中的函数$ q \，该功能是$ \ infty $。在Infinity $ Q^{'} \ lest（'}'} \ q^{'} \ left（'} \ left（\ weft（\ iffty \ right）： \ infty} {zq（z）} $是可逆操作员。事实证明，$ \ hat {q} $是总和$ \ hat {q} = \ hat {q} _ {1}+\ hat {q} _ {2}，\，\，\，\ hat {q} {q} _ {q} _ {i}满足$κ_{1}+κ_{2} =κ$的\ Mathcal {h} \ right）$。该分解使我们能够通过研究简单组件$ \ hat {q} _ {1} $和$ \ hat {q} _ {2} $来研究两个功能的属性，$ q $和$ q $和$ \ hat {q} $。
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 $\hat{Q}_{2}$.
