Frames of orbits of multiplication operators on Hardy spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918533821104128 |
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| author | Carando, Alejandra Aguilera adn Daniel |
| author_facet | Carando, Alejandra Aguilera adn Daniel |
| contents | We study frames for Hardy spaces generated by orbits of multiplication operators. We characterize the symbols $φ\in H^\infty(\mathbb{T}^N)$ for which the multiplication operator $M_φ$ admits a frame of orbits on $H^2(\mathbb{T}^N)$. We also show that, in this setting, the existence of a frame is equivalent to the existence of a Parseval frame. Moreover, for $N=1$ we prove that finitely many orbits suffice if and only if $φ$ is a finite Blaschke product. For $N > 1$, no finite collection of orbits can generate a frame, regardless of the symbol. We study the analogous problem for the adjoint operator $M_φ^*$. Our results extend to the infinite-dimensional torus $\mathbb{T}^\infty$ and, via Bohr's transform, to the Hardy space of Dirichlet series $\mathcal{H}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frames of orbits of multiplication operators on Hardy spaces Carando, Alejandra Aguilera adn Daniel Functional Analysis Classical Analysis and ODEs 42C15, 47B35, 30H10, 46E22 We study frames for Hardy spaces generated by orbits of multiplication operators. We characterize the symbols $φ\in H^\infty(\mathbb{T}^N)$ for which the multiplication operator $M_φ$ admits a frame of orbits on $H^2(\mathbb{T}^N)$. We also show that, in this setting, the existence of a frame is equivalent to the existence of a Parseval frame. Moreover, for $N=1$ we prove that finitely many orbits suffice if and only if $φ$ is a finite Blaschke product. For $N > 1$, no finite collection of orbits can generate a frame, regardless of the symbol. We study the analogous problem for the adjoint operator $M_φ^*$. Our results extend to the infinite-dimensional torus $\mathbb{T}^\infty$ and, via Bohr's transform, to the Hardy space of Dirichlet series $\mathcal{H}_2$. |
| title | Frames of orbits of multiplication operators on Hardy spaces |
| topic | Functional Analysis Classical Analysis and ODEs 42C15, 47B35, 30H10, 46E22 |
| url | https://arxiv.org/abs/2606.00912 |