We completely characterize non-periodic balanced weighted shifts $S_{ \lambdab}$ on rooted directed trees under a very mild assumption that $S_{ \lambdab}^{ *n}S_{ \lambdab}^n|_{ \ker S_{ \lambdab}^*}$ is invertible operator on $\ker S_{ \lambdab}^*$ for all $n \in \mathbb N$. This generalizes the previously established unitary equivalences for Bergman and Dirichlet type shifts associated with locally finite rooted directed trees. We also give a counter example to justify that the criteria obtained for non-periodic balanced weighted shifts is not necessary for eventually periodic balanced weighted shifts.
Categories: math.FA