Abstract
The weighted shifts are long known and form an important class of operators. One of generalisations of this class are weighted shifts on directed trees, where the linear order of coordinates in $$\ell ^2$$ is replaced by a more involved graph structure. In this paper we focus on the question of joint backward extending of a given family of weighted shifts on directed trees to a weighted shift on an enveloping directed tree that preserves subnormality or power hyponormality of considered operators. One of the main results shows that the existence of such a “joint backward extension” for a family of weighted shifts on directed trees depends only on the possibility of backward extending of single weighted shifts that are members of the family. We introduce a generalised framework of weighted shifts on directed forests (disjoint families of directed trees) which seems to be more convenient to work with. A characterisation of leafless directed forests on which all hyponormal weighted shifts are power hyponormal is given.
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