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