Closed ideals of operators on the Baernstein and Schreier spaces
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| Format: | Preprint |
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2024
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| _version_ | 1866916442078707712 |
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| author | Laustsen, Niels Jakob Smith, James |
| author_facet | Laustsen, Niels Jakob Smith, James |
| contents | We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces $B_p$ for $1<p<\infty$ and the Schreier spaces $S_p$ for $1\le p<\infty$. Our main conclusion is that there are $2^{\mathfrak{c}}$ many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also $2^{\mathfrak{c}}$ many closed ideals that contain projections of infinite rank.
Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the higher-order Schreier spaces play a key role in the proofs, as does the Johnson-Schechtman technique for constructing $2^{\mathfrak{c}}$ many closed ideals of operators on a Banach space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_12666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed ideals of operators on the Baernstein and Schreier spaces Laustsen, Niels Jakob Smith, James Functional Analysis 46H10, 47L10 (Primary) 46B03, 46B45, 47L20 (Secondary) We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces $B_p$ for $1<p<\infty$ and the Schreier spaces $S_p$ for $1\le p<\infty$. Our main conclusion is that there are $2^{\mathfrak{c}}$ many closed ideals that lie between the ideals of compact and strictly singular operators on each of these spaces, and also $2^{\mathfrak{c}}$ many closed ideals that contain projections of infinite rank. Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the higher-order Schreier spaces play a key role in the proofs, as does the Johnson-Schechtman technique for constructing $2^{\mathfrak{c}}$ many closed ideals of operators on a Banach space. |
| title | Closed ideals of operators on the Baernstein and Schreier spaces |
| topic | Functional Analysis 46H10, 47L10 (Primary) 46B03, 46B45, 47L20 (Secondary) |
| url | https://arxiv.org/abs/2410.12666 |