Finite rank operators in Jacobson radical ${\mathcal{R}}_{\mathcal{N}\otimes \mathcal{M}}$

Zhe, Dong

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 287-298.

Résumé

In this paper we investigate finite rank operators in the Jacobson radical ${\mathcal{R}}_{\mathcal{N}\otimes \mathcal{M}}$ of $\mathrm{A}lg(\mathcal{N}\otimes \mathcal{M})$, where $\mathcal{N}$, $\mathcal{M}$ are nests. Based on the concrete characterizations of rank one operators in $\mathrm{A}lg(\mathcal{N}\otimes \mathcal{M})$ 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 $\mathrm{A}lg(\mathcal{N}\otimes \mathcal{M})$ if and only if at least one of $\mathcal{N}$, $\mathcal{M}$ is continuous.

Détail

Classification : 47L35, 47L75
Mots-clés : Jacobson radical; finite rank operator