Factorization in Haar system Hardy spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909590353870848 |
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| author | Lechner, Richard Speckhofer, Thomas |
| author_facet | Lechner, Richard Speckhofer, Thomas |
| contents | A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I)_I$, either under a rearrangement-invariant norm $\|\cdot \|$ or under the associated square function norm \begin{equation*}
\Bigl\| \sum_Ia_Ih_I \Bigr\|_{*} = \Bigl\| \Bigl( \sum_I a_I^2 h_I^2 \Bigr)^{1/2} \Bigr\|. \end{equation*} Apart from $L^p$, $1\le p<\infty$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\ne C(Δ)$ ($C(Δ)$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \begin{equation*}
\mathcal{M}_Y = \{ T\in \mathcal{B}(Y) : I_Y \ne ATB\text{ for all } A, B\in \mathcal{B}(Y) \} \end{equation*} is the unique maximal ideal of the algebra $\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces $\ell^p(Y)$, $1\le p \leq \infty$, and use them to show that they are primary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10572 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Factorization in Haar system Hardy spaces Lechner, Richard Speckhofer, Thomas Functional Analysis 46B25, 46B09, 47A68, 47L20, 46E30, 30H10 A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I)_I$, either under a rearrangement-invariant norm $\|\cdot \|$ or under the associated square function norm \begin{equation*} \Bigl\| \sum_Ia_Ih_I \Bigr\|_{*} = \Bigl\| \Bigl( \sum_I a_I^2 h_I^2 \Bigr)^{1/2} \Bigr\|. \end{equation*} Apart from $L^p$, $1\le p<\infty$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\ne C(Δ)$ ($C(Δ)$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \begin{equation*} \mathcal{M}_Y = \{ T\in \mathcal{B}(Y) : I_Y \ne ATB\text{ for all } A, B\in \mathcal{B}(Y) \} \end{equation*} is the unique maximal ideal of the algebra $\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces $\ell^p(Y)$, $1\le p \leq \infty$, and use them to show that they are primary. |
| title | Factorization in Haar system Hardy spaces |
| topic | Functional Analysis 46B25, 46B09, 47A68, 47L20, 46E30, 30H10 |
| url | https://arxiv.org/abs/2310.10572 |