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Integral Equations and Operator Theory

, Volume 79, Issue 4, pp 567–577 | Cite as

Wandering Subspaces of the Bergman Space and the Dirichlet Space Over \({{\mathbb{D}^{n}}}\)

  • Arup Chattopadhyay
  • B. Krishna Das
  • Jaydeb SarkarEmail author
  • S. Sarkar
Article

Abstract

Doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc \({\mathbb{D}^n}\) (with \({n \geq 2}\)) are investigated. We show that for any non-empty subset \({\alpha=\{\alpha_1,\ldots,\alpha_k\}}\) of \({\{1,\ldots,n\}}\) and doubly commuting invariant subspace \({\mathcal{S}}\) of the Bergman space or the Dirichlet space over \({\mathbb{D}^n}\), restriction of the multiplication operator tuple on \({\mathcal{S}, M_{\alpha}|_\mathcal{S}:=(M_{z_{\alpha_1}}|_\mathcal{S},\ldots, M_{z_{\alpha_k}}|_\mathcal{S})}\), always possesses generating wandering subspace of the form
$$\bigcap_{i=1}^k(\mathcal{S}\ominus z_{\alpha_i}\mathcal{S})$$
.

Mathematics Subject Classification (2000)

47A13 47A15 47A20 47L99 

Keywords

Invariant subspace Beurling’s theorem Bergman space Dirichlet space Hardy space Doubly commuting invariant subspace 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Arup Chattopadhyay
    • 1
  • B. Krishna Das
    • 1
  • Jaydeb Sarkar
    • 1
    Email author
  • S. Sarkar
    • 2
  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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