Joint backward extension property for weighted shifts on directed trees

Weighted shifts on directed trees are a decade-old generalisation of classical shift operators in the sequence space ℓ2. In this paper we introduce joint backward extension property (JBEP) for classes weighted trees. If class satisfies JBEP, existence common within family rooted does not depend additional structure big tree (of fixed depth). We decide whether several have JBEP. For subnormal or power hyponormal is satisfied, while it fails completely hyperexpansive quasinormal. Nevertheless, some positive results extensions proven.