Wold-type decomposition for left-invertible weighted shifts on a rootless directed tree

Let $S_{\text{lambdab}}$ be a bounded left-invertible weighted shift on a rootless directed tree $\mathcal{T}=(V,\mathcal{E})\mathrm{.}$ We address the question of when $S_{\text{lambdab}}$ has Wold-type decomposition. We relate this problem to the convergence of the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\sum _{u\in G_{v,n}\mathrm{\backslash }{G}_{v,n-1}}(\frac{{\text{lambdab}}^{(n)}(u)}{{\text{lambdab}}^{(n)}(v)}{)}^2,}$ $v\in V,$ involving the moments ${\text{lambdab}}^{(n)}$ of $S_{\text{lambdab}}^{\ast }$, where $G_{v,n}=\text{childn}n\text{parentn}nv\mathrm{.}$ The main result of this paper characterizes all bounded left-invertible weighted shifts $S_{\text{lambdab}}$ on $\mathcal{T},$ which have Wold-type decomposition.