Invariant subspaces for certain tuples of operators with applications to reproducing kernel correspondences

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 $T=(T_1,\ldots,T_d)$ operators that are contractive and pure; that is $Φ_T(I)\leq I$ and $Φ_T^n(I)\searrow 0$ where $Φ_T(a)=Σ_i T_iaT_i^*.$ Here we generalize some of their results to commuting tuples $T$ satisfying similar conditions but for $Φ_T(a)=Σ_{α\in \mathbb{F}^+_d} x_{|α|}T_αaT_α^*$ where $\{x_k\}$ is a sequence of non negative numbers satisfying some natural conditions (where $T_α=T_{α(1)}\cdots T_{α(k)}$ for $k=|α|$). 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.

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