论文标题
通勤收缩的最小统一扩张
Minimal unitary dilations for commuting contractions
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 $，我们表明$（t_1，\ dots，dots，dots，t_n）$ dialist insometies $ nimeties $（v_1 $ triention $ triention $ t $ trimation $ triment $ trrimation $ trrimation $ thim \ dots $ thim， $ v = \ prod_ {i = 1}^n v_i $是$ t $的最小等距扩张，仅当$（t_1^*，\ dots，\ dots，t_n^*）$扩展到通勤异构体$（y__1，\ dots，y_n）$ the $ t^$ $ t^$ t^*$ t^*$ $ y = \ prod_ {i = 1}^n y_i $是$ t^*$的最小等距扩张。然后，我们证明了这种结果的类似物，即$（t_1，\ dots，t_n）$的单一扩张及其伴随。我们发现一种必要且充分的条件使得$（T_1，\ dots，t_n）$具有单一扩张$（W_1，\ dots，w_n）$，在$ t $的最小单位扩张空间，$ W = \ prod_ = \ prod_ = \ prod_ = \ prod_ = 1}^n w_i $是$ t $ t $的最小扩张。我们在SCH $ \ ddot {a} $ ffer和sz上展示了这种统一扩张的明确结构。 Nagy-foias最小的统一扩张空间为$ t $。另外，我们表明，当$ t $是$ c._0 $收缩时，相对较弱的假设对于存在这样的单一扩张是必要的，并且足够足够，即当$ {t^*}^n \ rightArrow 0 $ as $ n \ rightArrow $ n \ rightarrow \ rightarrow \ infty $。当$ t $是$ c._0 $收缩时，我们为$（t_1，\ dots，t_n）$构建了另一个单一扩张。
For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we show that $(T_1, \dots, T_n)$ dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. We construct a different unitary dilation for $(T_1, \dots , T_n)$ when $T$ is a $C._0$ contraction.
