An Analytic Model for Left-Invertible Weighted Shifts on Directed Trees

Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_{\lambda} \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_{\lambda}$ can be modelled as a multiplication operator $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on a disc centered at the origin, where $E:=\ker S^*_{\lambda}$. The reproducing kernel associated with $\mathscr H$ is multi-diagonal and of bandwidth $k_{\mathscr T}.$ Moreover, $\mathscr H$ admits an orthonormal basis consisting of polynomials in $z$ with at most $k_{\mathscr T}+1$ non-zero coefficients. As one of the applications of this model, we give a complete spectral picture of $S_{\lambda}.$ Unlike the case $\dim E = 1,$ the approximate point spectrum of $S_{\lambda}$ could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed trees with finite branching index.