The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples $$T=(T_1,\ldots ,T_d)$$ T = ( T 1 , … , T d ) of operators that are contractive and pure; that is $$\Phi _T(I)\le I$$ Φ T ( I ) ≤ I and $$\Phi _T^n(I)\searrow 0$$ Φ T n ( I ) ↘ 0 where $$\begin{aligned} \Phi _T(a)=\Sigma _i T_iaT_i^*. \end{aligned}$$ Φ T ( a ) = Σ i T i a T i ∗ . Here we generalize some of their results to commuting tuples T satisfying similar conditions but for $$\begin{aligned} \Phi _T(a)=\Sigma _{\alpha \in {\mathbb {F}}^+_d} x_{|\alpha |}T_{\alpha }aT_{\alpha }^* \end{aligned}$$ Φ T ( a ) = Σ α ∈ F d + x | α | T α a T α ∗ where $$\{x_k\}$$ { x k } is a sequence of non negative numbers satisfying some natural conditions (where $$T_{\alpha }=T_{\alpha (1)}\cdots T_{\alpha (k)}$$ T α = T α ( 1 ) ⋯ T α ( k ) for $$k=|\alpha |$$ k = | α | ). In fact, we deal with a more general situation where each $$x_k$$ x k is replaced by a $$d^k\times d^k$$ d k × d k matrix. We also apply these results to subspaces of certain reproducing kernel correspondences $$E_K$$ E K (associated with maps-valued kernels K ) that are invariant under the multipliers given by the coordinate functions.

中文翻译：

---

某些运算符元组的不变子空间与用于复制核对应的应用程序

由 Popescu、Muhly-Solel 和 Good 开发的用于研究由加权移位生成的代数的技术被应用于概括 Sarkar 和 Bhattacharjee-Eschmeier-Keshari-Sarkar 的关于膨胀和不变子空间的结果，用于交换运算符的元组。在该论文中，作者证明了 Beurling-Lax-Halmos 类型的结果，用于交换纯运算符的元组 $$T=(T_1,\ldots ,T_d)$$T = (T 1 , … , T d )；即 $$\Phi _T(I)\le I$$ Φ T ( I ) ≤ I 和 $$\Phi _T^n(I)\searrow 0$$ Φ T n ( I ) ↘ 0 其中 $$\开始{对齐} \Phi _T(a)=\Sigma _i T_iaT_i^*。\end{对齐}$$ Φ T ( a ) = Σ i T ia T i ∗ 。在这里，我们将他们的一些结果推广到满足类似条件的交换元组 T 但对于 $$\begin{aligned} \Phi _T(a)=\Sigma _{\alpha \in {\mathbb {F}}^+_d} x_ {|\alpha |}T_{\alpha }aT_{\alpha }^* \end{aligned}$$ Φ T ( a ) = Σ α ∈ F d + x | α | T α a T α ∗ 其中 $$\{x_k\}$$ { xk } 是满足某些自然条件的非负数序列（其中 $$T_{\alpha }=T_{\alpha (1)}\cdots T_{\alpha (k)}$$ T α = T α ( 1 ) ⋯ T α ( k ) for $$k=|\alpha |$$ k = | α | )。实际上，我们处理更一般的情况，其中每个 $$x_k$$ xk 被 $$d^k\times d^k$$ dk × dk 矩阵替换。我们还将这些结果应用于某些再现内核对应关系 $$E_K$$EK（与映射值内核 K 相关联）的子空间，这些子空间在坐标函数给定的乘数下是不变的。