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

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Reproducing Kernel
Invariant Subspace
Cardinality
Bergman Space
Zero
Unit Disk
Resolve
Multiplication
kernel
Operator

Keywords

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

Cite this

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title = "Reproducing kernels and invariant subspaces of the Bergman shift",
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.",
keywords = "Bergman shift, Bergman spaces, Bergman type kernels, Invariant subspaces, Reproducing kernels",
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Reproducing kernels and invariant subspaces of the Bergman shift. / Chailos, George.

In: Journal of Operator Theory, Vol. 51, No. 1, 12.2004, p. 181-200.

Research output: Contribution to journalArticle

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T1 - Reproducing kernels and invariant subspaces of the Bergman shift

AU - Chailos, George

PY - 2004/12

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N2 - 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.

AB - 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.

KW - Bergman shift

KW - Bergman spaces

KW - Bergman type kernels

KW - Invariant subspaces

KW - Reproducing kernels

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