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Finite rank operators in Jacobson radical $ℛ_{𝒩 \otimes ℳ}$

Zhe Dong — 2006

Czechoslovak Mathematical Journal

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

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