Ideals of regular operators on $l^{2}$

Abstract:

Let ${\mathcal {L}^r}$ be the Banach algebra (and Banach lattice) of all regular operators on ${l^{^2}}$, i.e. the algebra of all operators $A$ on ${l^2}$ which are given by a matrix $({a_{mn}})$ such that $(\left | {{a_{mn}}} \right |)$ defines a bounded operator $\left | A \right |$. We show that there exists exactly one nontrivial closed subspace of ${\mathcal {L}^r}$ which is both a lattice-ideal and an algebra-ideal of ${\mathcal {L}^r}$, namely the space ${\mathcal {K}^r} = \{ A \in {\mathcal {L}^r}:\left | A \right |{\text { is compact}}\}$. We also show that every nontrivial ideal in ${\mathcal {L}^r}$ is included in ${\mathcal {K}^r}$.