Doubly commuting submodules of the Hardy module over polydiscs

Volume 217 / 2013

Jaydeb Sarkar, Amol Sasane, Brett D. Wick Studia Mathematica 217 (2013), 179-192 MSC: Primary 46J15; Secondary 47A15, 30H05, 47A56. DOI: 10.4064/sm217-2-5


In this note we establish a vector-valued version of Beurling's theorem (the Lax–Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the “weak” completion problem in $H^\infty (\mathbb {D}^n)$.


  • Jaydeb SarkarIndian Statistical Institute
    Statistics and Mathematics Unit
    8th Mile, Mysore Road
    Bangalore, 560059, India
  • Amol SasaneMathematics Department
    London School of Economics
    Houghton Street
    London WC2A 2AE, U.K.
  • Brett D. WickSchool of Mathematics
    Georgia Institute of Technology
    686 Cherry Street
    Atlanta, GA 30332-0160, U.S.A.

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