Structure of the kernels associated with 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 double-struck D sign. 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. Here we study the boundary behaviour and some of the basic properties of the kernel lλ. Among other things, we show that if the lower zero set of M, Z(M), is nonempty, the kernel lλ for fixed λ ∈ double-struck D sign has a meromorphic continuation across double-struck T sign/Z(M), where double-struck T struck is the unit circle. Furthermore we consider some special types of kernels lλ and by studying their structure we obtain information for the invariant subspaces related to them. Lastly, and after introducing the general vector valued setting, we discuss some analogous results for the case of ⊕ La2(double-struck D sign;), where m is a positive integer.

Original languageEnglish
Pages (from-to)445-457
Number of pages13
JournalBulletin of the Australian Mathematical Society
Volume67
Issue number3
Publication statusPublished - Jun 2003

Keywords

  • Reproducing kernels
  • Hilbert Spaces
  • Invariant subspaces
  • Bergmann spaces

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