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

Abstract

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