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.