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


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 


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aleman A., Richter S., Sundberg C.: Beurling’s theorem for the Bergman space. Acta. Math. 177, 275–310 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beurling A.: On two problems concerning linear transformations in Hilbert space. Acta. Math. 81, 239–255 (1949)CrossRefzbMATHGoogle Scholar
  4. 4.
    Duren, P., Schuster, A.: Bergman spaces. Mathematical Surveys and Monographs, vol. 100. American Mathematical Society ProvidenceGoogle Scholar
  5. 5.
    Halmos P.R.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Mandrekar V.: The validity of Beurling theorems in polydiscs. Proc. Amer. Math. Soc. 103, 145–148 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Redett D., Tung J.: Invariant subspaces in Bergman space over the bidisc. Proc. Am. Math. Soc. 138, 2425–2430 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Richter S.: Invariant subspaces of the Dirichlet shift. J. Reine Angew. Math. 386, 205–220 (1988)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Rudin W.: Function theory in polydiscs. Benjamin, New York (1969)zbMATHGoogle Scholar
  10. 10.
    Shimorin S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Sarkar J., Sasane A., Wick B.: Doubly commuting submodules of the Hardy module over polydiscs. Studia Math. 217(2), 179–192 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wold H.: A study in the analysis of stationary time series. Almquist and Wiksell, Uppsala (1938)zbMATHGoogle Scholar

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

Personalised recommendations