Invariant subspaces for finite index shifts in Hardy spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912903829913600 |
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| author | Bracci, Filippo Gallardo-Gutiérrez, Eva A. |
| author_facet | Bracci, Filippo Gallardo-Gutiérrez, Eva A. |
| contents | Let $\mathbb H$ be the finite direct sums of $H^2(\mathbb D)$. In this paper, we give a characterization of the closed subspaces of $\mathbb H$ which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of $H^2(\mathbb D)$ under ``determinantal operators'' from $\mathbb H$ to $H^2(\mathbb D)$, that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every contraction with finite defect has a nontrivial closed invariant subspace. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant subspaces for finite index shifts in Hardy spaces Bracci, Filippo Gallardo-Gutiérrez, Eva A. Functional Analysis Complex Variables Operator Algebras Let $\mathbb H$ be the finite direct sums of $H^2(\mathbb D)$. In this paper, we give a characterization of the closed subspaces of $\mathbb H$ which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of $H^2(\mathbb D)$ under ``determinantal operators'' from $\mathbb H$ to $H^2(\mathbb D)$, that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every contraction with finite defect has a nontrivial closed invariant subspace. |
| title | Invariant subspaces for finite index shifts in Hardy spaces |
| topic | Functional Analysis Complex Variables Operator Algebras |
| url | https://arxiv.org/abs/2411.01933 |