标题： 纯压缩乘子的一些再生核希尔伯特空间及其应用 显示英文标题

标题： Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

摘要： 在希尔伯特空间$\mathcal{H}$上的压缩$T$被称为纯的，如果序列$\lbrace T^{*n} \rbrace_{n}$在强算子拓扑中收敛到$0$。 在本文中，我们证明对于与某些可处理的交换算子元组$X = (X_1,\ldots,X_n)$在$\mathcal{H}$上交换的收缩算子$T$，以下陈述是等价的：(i)$T$是$\mathcal{H}$上的纯收缩算子，(ii) 压缩算子$P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$是纯收缩算子，其中$\mathcal{W}(X)$是对应于元组$X$的漫游子空间。 一个向量值再生核希尔伯特空间（rkHs）的算子值乘子$\Phi$被称为纯压缩的，如果相关的乘法算子$M_{\Phi}$是一个纯压缩算子。 利用上述结果，我们发现几个向量值 rkHs 在多圆盘$\mathbb{D}^n$以及单位球$\mathbb{B}_n$上的算子值乘子$\Phi(\textbf{z})$以及在$\mathbb{C}^n$中的纯压缩性当且仅当$\Phi(0)$在基础希尔伯特空间上是一个纯压缩算子。 该列表包括 Hardy 空间、Bergman 空间和 Drury-Arveson 空间。 最后，我们展示了我们对多圆盘相关纯压缩乘子的特征的一些应用。 显示英文摘要

摘要： A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operator topology. In this article, we prove that for contractions $T$, which commute with certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued multiplier $\Phi$ of a vector-valued reproducing kernel Hilbert space (rkHs) is said to be pure contractive if the associated multiplication operator $M_{\Phi}$ is a pure contraction. Using the above result, we find that operator-valued mulitpliers $\Phi(\textbf{z})$ of several vector-valued rkHs's on the polydisc $\mathbb{D}^n$ as well as the unit ball $\mathbb{B}_n$ in $\mathbb{C}^n$ are pure contractive if and only if $\Phi(0)$ is a pure contraction on the underlying Hilbert space. The list includes Hardy, Bergman and Drury-Arveson spaces. Finally, we present some applications of our characterization of pure contractive multipliers associated with the polydisc.