An operator ideal $\mathfrak J$ is a class of continuous operators. Namely, for every pair of complex Banach spaces, $\mathfrak X,\mathfrak Y$, we have that $\mathfrak J(\mathfrak X,\mathfrak Y) \subseteq \mathfrak L(\mathfrak X,\mathfrak Y)$ is a closed two-sided ideal, which means \begin{align*} 1.& \ \ \ A,B\in \mathfrak J(\mathfrak X,\mathfrak Y) \Rightarrow A+B \in \mathfrak J(\mathfrak X,\mathfrak Y), \\ 2.& \ \ \ \mathfrak L(\mathfrak W,\mathfrak X)\mathfrak J(\mathfrak X,\mathfrak Y)\mathfrak L(\mathfrak Y,\mathfrak Z) \subseteq \mathfrak J(\mathfrak W,\mathfrak Z),\ \textrm{and} \\ 3.& \ \ \ \mathfrak J(\mathfrak X,\mathfrak Y) \supseteq \mathfrak F(\mathfrak X,\mathfrak Y), \ \textrm{the finite-rank operators}. \end{align*}

Now for my question: Let $\mathfrak X$ be any complex Banach space and suppose $J$ is a non-trivial closed two-sided ideal of $\mathfrak L(\mathfrak X)$. Can you always find an operator ideal $\mathfrak J$ such that $\mathfrak J(\mathfrak X,\mathfrak X) = J$?

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