Sequence-singular operators

In this paper we study ℓpℓp-related collections of operators introduced by Beanland and Freeman in [6], on the subject of forming operator ideals. We show that these collections are not always closed under addition, and hence do not form operator ideals. Nevertheless, they allow us to construct an uncountable chain of closed ideals in each of the operator algebras L(ℓ1⊕ℓq)L(ℓ1⊕ℓq), 1