Weighted shifts on directed forests and hyponormality

In a 2012 paper, Jabłoński, Jung and Stochel introduced weighted shifts on directed trees, a generalisation of the well-known weighted shift operators on $\ell^2$. In the last decade this class has proven itself handy for finding counterexamples in operator theory. Properties of the underlying graph structure had essential influence on these operators. It appears that a slight generalisation of the class, namely weighted shifts on directed forests, shows even deeper relations between graph theory and operator theory. Several operations on directed forests have their natural operator-theoretic counterparts. This paper is meant to present advantages of the directed forest approach. As an application of the interrelation between graphs and operators we provide a full characterisation of directed forests on which every hyponormal bounded weighted shift is power hyponormal.