Closed ideals in \(\mathcal{L}(X)\) and \(\mathcal{L}(X^)\) when \(X\) contains certain copies of \(\ell_p\) and \(c_0\)

Suppose \(X\) is a real or complexified Banach space containing a complemented copy of \(\ell_p\), \(p\in(1,2)\), and a copy (not necessarily complemented) of either \(\ell_q\), \(q\in(p,\infty)\), or \(c_0\). Then \(\mathcal{L}(X)\) and \(\mathcal{L}(X^*)\) each admit continuum many closed ideals....

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Bibliographic Details
Published in:arXiv.org 2015-07
Main Author: Wallis, Ben
Format: Article
Language:English
Subjects:
Banach spaces
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Summary:Suppose \(X\) is a real or complexified Banach space containing a complemented copy of \(\ell_p\), \(p\in(1,2)\), and a copy (not necessarily complemented) of either \(\ell_q\), \(q\in(p,\infty)\), or \(c_0\). Then \(\mathcal{L}(X)\) and \(\mathcal{L}(X^*)\) each admit continuum many closed ideals. If in addition \(q\geq p'\), \(\frac{1}{p}+\frac{1}{p'}=1\), then the closed ideals of \(\mathcal{L}(X)\) and \(\mathcal{L}(X^*)\) each fail to be linearly ordered. We obtain additional results in the special cases of \(\mathcal{L}(\ell_1\oplus\ell_q)\) and \(\mathcal{L}(\ell_p\oplus c_0)\), \(1
ISSN:2331-8422