论文标题
多盘的最小等距扩张和操作员模型
Minimal isometric dilations and operator models for the polydisc
论文作者
论文摘要
对于通勤收缩，$ t_1，\ dots，t_n $作用于希尔伯特太空上的$ \ mathcal h $，$ t = \ prod_ {i = 1}^n t_i $，我们发现$（t_1，\ dots，t_n）的必要条件，以$（t_1，t_n）$ imimities $ nimemeties $ nim inim $ nim inim inim $ nim nim nim nim inim $ nim inim $ nim nim nim nim nim nim nim inim（v_1）扩张空间$ t $，其中$ v = \ prod_ {i = 1}^nv_i $是$ t $的最小等距扩张。我们构建两个sch $ \ ddot {a} $ ffer和sz。 $（t_1，\ dots，t_n）$的Nagy-foias类型的等距扩张，$ t $。另外，当产品$ t $是$ c._0 $收缩时，构建了不同的扩张，即$ {t^*}^n \ rightarrow 0 $ as $ n \ rightarrow \ rightArrow \ infty $。由于这些扩张定理，我们在矢量强烈空间上的乘法运算符方面获得了$（t_1，\ dots，t_n）$的不同功能模型。关于我们的模型的一个值得注意的事实是，乘数是一个变量中的分析函数。当$ t $是$ c._0 $收缩时，扩张会导致$ t $的条件分解。已经构建了几个示例。
For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for $(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different dilation is constructed when the product $T$ is a $C._0$ contraction, that is ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed.
