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Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces
| Content Provider | Semantic Scholar |
|---|---|
| Author | Barbian, Christoph |
| Copyright Year | 2008 |
| Abstract | Abstract.By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space $$H^2({\mathbb{D}})$$ on the unit disk can be represented as $$P_M = M_\phi M_\phi^*$$ where Φ is an inner multiplier of $$H^2({\mathbb{D}})$$. This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers $$\phi_1, \phi_2$$ such that $$P_M = M_{\phi_1} M_{\phi_1}^* - M_{\phi_2} M_{\phi_2}^*$$ and $$M = {\rm ran}\, M_{\phi_1}$$. We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules $${\mathcal{H}}$$, every finite-codimensional submodule M can be written as $$M = \sum^r_ {i=1} p_i \cdot {\mathcal{H}}$$ with suitable polynomials pi. |
| Starting Page | 299 |
| Ending Page | 323 |
| Page Count | 25 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00020-008-1590-9 |
| Volume Number | 61 |
| Alternate Webpage(s) | https://publikationen.sulb.uni-saarland.de/bitstream/20.500.11880/26109/1/Barbian.pdf |
| Alternate Webpage(s) | http://www.math.uni-sb.de/service/preprints/preprint167.pdf |
| Alternate Webpage(s) | https://www.math.uni-sb.de/PREPRINTS/preprint167.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00020-008-1590-9 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
