Invariance and near invariance for non-cyclic shift semigroups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914446795866112 |
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| author | Liang, Yuxia Partington, Jonathan R. |
| author_facet | Liang, Yuxia Partington, Jonathan R. |
| contents | This paper characterises the subspaces of $H^2(\mathbb D)$ simultaneously invariant under $S^2 $ and $S^{2k+1}$, where $S$ is the unilateral shift, then further identifies the subspaces that are nearly invariant under both $(S^2)^*$ and $(S^{2k+1})^*$ for $k\geq 1$. More generally, the simultaneously (nearly) invariant subspaces with respect to $(S^m)^*$ and $(S^{km+γ})^*$ are characterised for $m\geq 3$, $k\geq 1$ and $γ\in \{1,2,\cdots, m-1\},$ which leads to a description of (nearly) invariant subspaces with respect to higher order shifts. Finally, the corresponding results for Toeplitz operators induced by a Blaschke product are presented. Techniques used include a refinement of Hitt's algorithm, the Beurling--Lax theorem, and matrices of analytic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariance and near invariance for non-cyclic shift semigroups Liang, Yuxia Partington, Jonathan R. Functional Analysis Complex Variables 47B38, 47A15, 30H10 This paper characterises the subspaces of $H^2(\mathbb D)$ simultaneously invariant under $S^2 $ and $S^{2k+1}$, where $S$ is the unilateral shift, then further identifies the subspaces that are nearly invariant under both $(S^2)^*$ and $(S^{2k+1})^*$ for $k\geq 1$. More generally, the simultaneously (nearly) invariant subspaces with respect to $(S^m)^*$ and $(S^{km+γ})^*$ are characterised for $m\geq 3$, $k\geq 1$ and $γ\in \{1,2,\cdots, m-1\},$ which leads to a description of (nearly) invariant subspaces with respect to higher order shifts. Finally, the corresponding results for Toeplitz operators induced by a Blaschke product are presented. Techniques used include a refinement of Hitt's algorithm, the Beurling--Lax theorem, and matrices of analytic functions. |
| title | Invariance and near invariance for non-cyclic shift semigroups |
| topic | Functional Analysis Complex Variables 47B38, 47A15, 30H10 |
| url | https://arxiv.org/abs/2408.08659 |