We formulate general conditions which imply that ${\mathcal L}(X,Y)$ , the space of operators from a Banach space X to a Banach space Y, has $2^{{\mathfrak {c}}}$ closed ideals, where ${\mathfrak {c}}$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ is exactly $2^{{\mathfrak {c}}}$ for all $1<p<q<\infty $ .

中文翻译：

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算子空间有 2𝔠 封闭理想的 Banach 空间

我们制定一般条件，这意味着 ${\mathcal L}(X,Y)$ , 来自 Banach 空间的算子空间X到 Banach 空间是， 已 $2^{{\mathfrak {c}}}$ 封闭的理想，其中 ${\mathfrak {c}}$ 是连续统的基数。这些结果适用于经典序列空间和 Tsirelson 型空间。特别地，我们证明了闭理想集的基数 ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ 正是 $2^{{\mathfrak {c}}}$ 对所有人 $1<p<q<\infty $ .