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.