Reproducing kernels and invariant subspaces of the Bergman shift

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Abstract

In this article we consider index 1 invariant subspaces M of the operator of multiplication by ζ(z) = z, Mζ, on the Bergman space La2(double struck D sign) of the unit disc. It turns out that there is a positive sesquianalytic kernel lλ defined on double struck D sign × double struck D sign which determines M uniquely. We set σ(M*ζ|M) to be the spectrum of M*ζ restricted to M, and we consider a conjecture due to Hedenmalm which states that if M ≠ La 2(double struck D sign), then rank lλ equals the cardinality of σ(M*ζ|M). In this direction we show that cardinality (σ(M*ζ|M ) ∩ double struck D sign ≤ rank lλ ≤ cardinality σ(M*ζ|M) and furthermore, we resolve the conjecture in the case of zero based invariant subspaces. Moreover, we describe the structure of lλ for finite zero based invariant subspaces.

Original languageEnglish
Pages (from-to)181-200
Number of pages20
JournalJournal of Operator Theory
Volume51
Issue number1
Publication statusPublished - Dec 2004

Keywords

  • Bergman shift
  • Bergman spaces
  • Bergman type kernels
  • Invariant subspaces
  • Reproducing kernels

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