Invariant Subspaces for Certain Tuples of Operators with Applications to Reproducing Kernel Correspondences

Abstract

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 $T=(T_1,\ldots ,T_d)$ of operators that are contractive and pure; that is $\Phi _T(I)\le I$ and $\Phi _T^n(I)\searrow 0$ where

$$\begin{aligned} \Phi _T(a)=\Sigma _i T_iaT_i^*. \end{aligned}$$

Here we generalize some of their results to commuting tuples T satisfying similar conditions but for

$$\begin{aligned} \Phi _T(a)=\Sigma _{\alpha \in {\mathbb {F}}^+_d} x_{|\alpha |}T_{\alpha }aT_{\alpha }^* \end{aligned}$$

where $\{x_k\}$ is a sequence of non negative numbers satisfying some natural conditions (where $T_{\alpha }=T_{\alpha (1)}\cdots T_{\alpha (k)}$ for $k=|\alpha |$). In fact, we deal with a more general situation where each $x_k$ is replaced by a $d^k\times d^k$ matrix. We also apply these results to subspaces of certain reproducing kernel correspondences $E_K$ (associated with maps-valued kernels K) that are invariant under the multipliers given by the coordinate functions.