Geodesic
Finite rank operators in Jacobson radical ${\scr R}\sb{{\scr N}\otimes{\scr M}}$
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 287-298 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 47L35, 47L75
Keywords: Jacobson radical; finite rank operator 
@article{CMJ_2006_56_2_a1,
     author = {Zhe, Dong},
     title = {Finite rank operators in {Jacobson} radical ${\scr R}\sb{{\scr N}\otimes{\scr M}}$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {287--298},
     year = {2006},
     volume = {56},
     number = {2},
     mrnumber = {2291737},
     zbl = {1164.47398},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_2_a1/}
}
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Zhe, Dong. Finite rank operators in Jacobson radical ${\scr R}\sb{{\scr N}\otimes{\scr M}}$. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 287-298. http://geodesic.mathdoc.fr/item/CMJ_2006_56_2_a1/