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 $T=(T_1,\dots ,T_d)$ operators that are contractive and pure; that is ${\Phi }_T(I)\le I$ and ${\Phi }_T^n(I)\searrow 0$ where $${\Phi }_T(a)={\Sigma }_i{T}_{i}a{T}_i^{\ast }.$$ Here we generalize some of their results to commuting tuples $T$ satisfying similar conditions but for $${\Phi }_T(a)={\Sigma }_{\alpha \in {\mathbb{F}}_d^{+}}{x}_{|\alpha |}{T}_{\alpha }a{T}_{\alpha }^{\ast }$$ 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.