Bergman内函数与$m$ -超收缩
Bergman inner functions and $m$-hypercontractions
摘要：
设$H_m(\mathbb B,\mathcal D)$是具有再生核$K_m(z,w) = (1-\langle z,w\rangle)^{-m}1_{\mathcal D}$的$\mathcal D$值泛函Hilbert空间. $K_m$内部函数根据定义是算子值解析函数$W: \mathbb B \rightarrow L(\mathcal E, \mathcal D)$，使得对于所有$x \in \mathcal E$为$\|Wx\|_{H_m(\mathbb B,\mathcal D)} = \|x\|$，对于所有$α\in \mathbb N^n \setminus \{0\}$为$(W\mathcal E) \perp M_z^α(W\mathcal E)$。我们证明了$K_m$ -内函数是$W(z) = D + C \sum^m_{k=1}(1 - ZT^*)^{-k}ZB$形式的函数，其中$T \in L(H)^n$是一个纯$m$ -超压缩，而算子$T^*, B, C,D$构成一个满足适当条件的$2 \times 2$ -算子矩阵.因此，我们将Olofsson在单位圆盘上证明的结果推广到单位球$\mathbb B \subset \mathbb C^n$的情形.
英文摘要：
Let $H_m(\mathbb B,\mathcal D)$ be the $\mathcal D$-valued functional Hilbert space with reproducing kernel $K_m(z,w) = (1-\langle z,w\rangle)^{-m}1_{\mathcal D}$. A $K_m$-inner function is by definition an operator-valued analytic function $W: \mathbb B \rightarrow L(\mathcal E, \mathcal D)$ such that $\|Wx\|_{H_m(\mathbb B,\mathcal D)} = \|x\|$ for all $x \in \mathcal E$ and $(W\mathcal E) \perp M_z^α(W\mathcal E)$ for all $α\in \mathbb N^n \setminus \{0\}$. We show that the $K_m$-inner functions are precisely the functions of the form $W(z) = D + C \sum^m_{k=1}(1 - ZT^*)^{-k}ZB$, where $T \in L(H)^n$ is a pure $m$-hypercontraction and the operators $T^*, B, C,D$ form a $2 \times 2$-operator matrix satisfying suitable conditions. Thus we extend results proved by Olofsson on the unit disc to the case of the unit ball $\mathbb B \subset \mathbb C^n$.
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