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

TL;DR

This paper explores the structure of weighted shifts on generalized directed semi-trees, establishing their connection to multiplication operators on certain Hilbert spaces of holomorphic functions, with concrete examples illustrating the theory.

Contribution

It introduces generalized directed semi-trees, constructs weighted shifts on them, and links these operators to multiplication operators on reproducing kernel Hilbert spaces, providing new insights into their analytic structure.

Findings

Weighted shifts on generalized directed semi-trees can be modeled as multiplication operators.

The associated Hilbert spaces are reproducing kernel Hilbert spaces of holomorphic functions.

Two classes of examples demonstrate the intrinsic identification of these operators as weighted shifts.

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 $\mathscr{H}^2(\beta)$ of formal power series. Under certain conditions, $\mathscr{H}^2(\beta)$ 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 $\mathscr{H}^2(\beta)$ 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.