On analytic structure of weighted shifts on generalized directed semi-trees

Abstract

Inspired by natural classes of examples, we define generalized directed semi-tree and construct weighted shifts on the generalized directed semi-trees. Given an $n$-tuple of directed directed semi-trees with certain properties, we associate an $n$-tuple of multiplication operators on a Hilbert space ${\mathcal{H}}^2(\beta )$ of formal power series. Under certain conditions, ${\mathcal{H}}^2(\beta )$ turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in ${\mathbb{C}}^n$ and the $n$-tuple of multiplication operators on ${\mathcal{H}}^2(\beta )$ is unitarily equivalent to an $n$-tuple of weighted shifts on the generalized directed semi-trees. Finally, we exhibit two classes of examples of $n$-tuple of operators which can be intrinsically identified as weighted shifts on generalized directed semi-trees.