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.

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Correspondence to Baruch Solel.

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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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Keywords

  • Invariant subspace
  • Beurling Lax Halmos theorem
  • Commuting tuple of operators
  • Kernel
  • Correspondence
  • Reproducing kernel

Mathematics Subject Classification

  • 30H05
  • 46E22
  • 46L08
  • 47A13
  • 47A15
  • 47A20
  • 47B32
  • 47B38