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Jacobson radical; finite rank operator
In this paper we investigate finite rank operators in the Jacobson radical $\mathcal R_{\mathcal N\otimes \mathcal M}$ of $\mathop {\mathrm Alg}(\mathcal N\otimes \mathcal M)$, where $\mathcal N$, $\mathcal M$ are nests. Based on the concrete characterizations of rank one operators in $\mathop {\mathrm Alg}(\mathcal N\otimes \mathcal M)$ and $\mathcal R_{\mathcal N\otimes \mathcal M}$, we obtain that each finite rank operator in $\mathcal R_{\mathcal N\otimes \mathcal M}$ can be written as a finite sum of rank one operators in $\mathcal R_{\mathcal N\otimes \mathcal M}$ and the weak closure of $\mathcal R_{\mathcal N\otimes \mathcal M}$ equals $\mathop {\mathrm Alg}({\mathcal N\otimes \mathcal M})$ if and only if at least one of $\mathcal N$, $\mathcal M$ is continuous.
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