### 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 |

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### Keywords

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

### Cite this

*Journal of Operator Theory*,

*51*(1), 181-200.

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*Journal of Operator Theory*, vol. 51, no. 1, pp. 181-200.

**Reproducing kernels and invariant subspaces of the Bergman shift.** / Chailos, George.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Reproducing kernels and invariant subspaces of the Bergman shift

AU - Chailos, George

PY - 2004/12

Y1 - 2004/12

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

UR - http://www.scopus.com/inward/record.url?scp=2442528928&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:2442528928

VL - 51

SP - 181

EP - 200

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 1

ER -