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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.20346 |
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| _version_ | 1866910505024618496 |
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| author | Bampouras, Konstantinos Brevig, Ole Fredrik |
| author_facet | Bampouras, Konstantinos Brevig, Ole Fredrik |
| contents | Let $H$ be a Hilbert space that can be embedded as a dense subspace of a Banach space $X$ such that the norm of the embedding is equal to $1$. We consider the following statements for a nonzero vector $φ$ in $H$:
(A) $\|φ\|_X = \|φ\|_H$.
(H) $\|φ+f\|_X \geq \|φ\|_X$ for every $f$ in $H$ such that $\langle f, φ\rangle =0$. We use duality arguments to establish that (A) $\implies$ (H), before turning our attention to the special case when the Hilbert space in question is the Hardy space $H^2(\mathbb{T}^d)$ and the Banach space is either the Hardy space $H^1(\mathbb{T}^d)$ or the weak product space $H^2(\mathbb{T}^d) \odot H^2(\mathbb{T}^d)$. If $d=1$, then the two Banach spaces are equal and it is known that (H) $\implies$ (A). If $d\geq2$, then the Banach spaces do not coincide and a case study of the polynomials $φ_α(z) = z_1^2 + αz_1 z_2 + z_2^2$ for $α\geq0$ illustrates that the statements (A) and (H) for the two Banach spaces describe four distinct sets of functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20346 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Norm attaining vectors and Hilbert points Bampouras, Konstantinos Brevig, Ole Fredrik Functional Analysis Let $H$ be a Hilbert space that can be embedded as a dense subspace of a Banach space $X$ such that the norm of the embedding is equal to $1$. We consider the following statements for a nonzero vector $φ$ in $H$: (A) $\|φ\|_X = \|φ\|_H$. (H) $\|φ+f\|_X \geq \|φ\|_X$ for every $f$ in $H$ such that $\langle f, φ\rangle =0$. We use duality arguments to establish that (A) $\implies$ (H), before turning our attention to the special case when the Hilbert space in question is the Hardy space $H^2(\mathbb{T}^d)$ and the Banach space is either the Hardy space $H^1(\mathbb{T}^d)$ or the weak product space $H^2(\mathbb{T}^d) \odot H^2(\mathbb{T}^d)$. If $d=1$, then the two Banach spaces are equal and it is known that (H) $\implies$ (A). If $d\geq2$, then the Banach spaces do not coincide and a case study of the polynomials $φ_α(z) = z_1^2 + αz_1 z_2 + z_2^2$ for $α\geq0$ illustrates that the statements (A) and (H) for the two Banach spaces describe four distinct sets of functions. |
| title | Norm attaining vectors and Hilbert points |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2310.20346 |