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
Here we generalize some of their results to commuting tuples T satisfying similar conditions but for
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.
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Solel, B. Invariant Subspaces for Certain Tuples of Operators with Applications to Reproducing Kernel Correspondences. Integr. Equ. Oper. Theory 92, 38 (2020). https://doi.org/10.1007/s00020-020-02596-3
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DOI: https://doi.org/10.1007/s00020-020-02596-3
Keywords
- Invariant subspace
- Beurling Lax Halmos theorem
- Commuting tuple of operators
- Kernel
- Correspondence
- Reproducing kernel