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

Inspired by natural classes of examples, we define generalized directed semi-trees and construct weighted shifts on them. Given an n-tuple of generalized directed semi-trees with certain properties, we associate an n-tuple of multiplication operators on a Hilbert space H 2 (β) of formal power series. Under certain conditions, H 2 (β) turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in C n and the n-tuple of multiplication operators on H 2 (β) is unitarily equivalent to an n-tuple of weighted shifts on generalized directed semi-trees. Finally, we exhibit two classes of examples of n-tuples of operators, which can be intrinsically identified as weighted shifts on generalized directed semi-trees. [ABSTRACT FROM AUTHOR]