### 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 double-struck D sign. 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. Here we study the boundary behaviour and some of the basic properties of the kernel l_{λ}. Among other things, we show that if the lower zero set of M, Z(M), is nonempty, the kernel l_{λ} 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 l_{λ} 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 ⊕ L_{a}^{2}(double-struck D sign;), where m is a positive integer.

Original language | English |
---|---|

Pages (from-to) | 445-457 |

Number of pages | 13 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 67 |

Issue number | 3 |

Publication status | Published - Jun 2003 |

### Fingerprint

### Keywords

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

### Cite this

}

*Bulletin of the Australian Mathematical Society*, vol. 67, no. 3, pp. 445-457.

**Structure of the kernels associated with invariant subspaces of the Bergman shift.** / Chailos, George.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Structure of the kernels associated with invariant subspaces of the Bergman shift

AU - Chailos, George

PY - 2003/6

Y1 - 2003/6

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 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.

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 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.

KW - Reproducing kernels

KW - Hilbert Spaces

KW - Invariant subspaces

KW - Bergmann spaces

UR - http://www.scopus.com/inward/record.url?scp=30244533252&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:30244533252

VL - 67

SP - 445

EP - 457

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 3

ER -