An analytic model for left-invertible weighted shifts on directed trees

Abstract

Let be a rooted directed tree with finite branching index , and let be a left-invertible weighted shift on . We show that can be modelled as a multiplication operator on a reproducing kernel Hilbert space of -valued holomorphic functions on a disc centred at the origin, where . The reproducing kernel associated with is multi-diagonal and of bandwidth Moreover, admits an orthonormal basis consisting of polynomials in with at most non-zero coefficients. As one of the applications of this model, we give a spectral picture of Unlike the case , the approximate point spectrum of could be disconnected. We also obtain an analytic model for left-invertible weighted shifts on rootless directed tree with finite branching index.