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 language | English |
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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 |
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Keywords
- Reproducing kernels
- Hilbert Spaces
- Invariant subspaces
- Bergmann spaces
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Structure of the kernels associated with invariant subspaces of the Bergman shift. / Chailos, George.
In: Bulletin of the Australian Mathematical Society, Vol. 67, No. 3, 06.2003, p. 445-457.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 -