Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914176762380288 |
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| author | Laustsen, Niels Jakob Wirzenius, Henrik |
| author_facet | Laustsen, Niels Jakob Wirzenius, Henrik |
| contents | Let $X$ be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces $B_p$ for $1<p<\infty$; (ii) the $p$-convexified Schreier spaces $S_p$ for $1\le p<\infty$; (iii) the sequence spaces $\ell_p$ for $1\le p<\infty$ (and $c_0$). We show that the quotient algebra of strictly singular by compact operators on $X$ is nilpotent; that is, there is a natural number $k$, dependent only on the collections of direct summands from each of the three families, such that:
- every composition of $k+1$ strictly singular operators on $X$ is compact;
- there are $k$ strictly singular operators on $X$ whose composition is not compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces Laustsen, Niels Jakob Wirzenius, Henrik Functional Analysis 16N40, 46B45, 47L10 (primary) 46B03, 46H10, 47L20 (secondary) Let $X$ be the direct sum of finitely many Banach spaces chosen from the following three families: (i) the Baernstein spaces $B_p$ for $1<p<\infty$; (ii) the $p$-convexified Schreier spaces $S_p$ for $1\le p<\infty$; (iii) the sequence spaces $\ell_p$ for $1\le p<\infty$ (and $c_0$). We show that the quotient algebra of strictly singular by compact operators on $X$ is nilpotent; that is, there is a natural number $k$, dependent only on the collections of direct summands from each of the three families, such that: - every composition of $k+1$ strictly singular operators on $X$ is compact; - there are $k$ strictly singular operators on $X$ whose composition is not compact. |
| title | Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces |
| topic | Functional Analysis 16N40, 46B45, 47L10 (primary) 46B03, 46H10, 47L20 (secondary) |
| url | https://arxiv.org/abs/2509.02405 |