### Abstract

In this article we consider index 1 invariant subspaces M of the operator of multiplication by ζ(z) = z, M_{ζ}, on the Bergman space L_{a}^{2}(double struck D sign) of the unit disc. It turns out that there is a positive sesquianalytic kernel l_{λ} 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 ≠ L_{a} ^{2}(double struck D sign), then rank l_{λ} equals the cardinality of σ(M*_{ζ}|M^{⊥}). In this direction we show that cardinality (σ(M*_{ζ}|M ^{⊥}) ∩ double struck D sign ≤ rank l_{λ} ≤ 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 l_{λ} for finite zero based invariant subspaces.

Original language | English |
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Pages (from-to) | 181-200 |

Number of pages | 20 |

Journal | Journal of Operator Theory |

Volume | 51 |

Issue number | 1 |

Publication status | Published - Dec 2004 |

### Keywords

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

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## Cite this

*Journal of Operator Theory*,

*51*(1), 181-200.