Unbounded subnormal weighted shifts on directed trees

TL;DR

This paper introduces a new approximation-based method to verify subnormality of unbounded weighted shifts on directed trees, expanding understanding of their properties and conditions for subnormality.

Contribution

It develops a novel approach using probability measures and extends Lambert's bounded operator results to certain unbounded shifts on directed trees.

Findings

Established sufficient conditions for subnormality of unbounded weighted shifts.

Linked subnormality to determinate Stieltjes moment sequences.

Extended Lambert's characterization to unbounded operators with quasi-analytic vectors.

Abstract

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established. An approach to this issue via consistent systems of probability measures is invented. The role played by determinate Stieltjes moment sequences is elucidated. Lambert's characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi-analytic vectors, which is a new phenomenon in this area. The cases of classical weighted shifts and weighted shifts on leafless directed trees with one branching vertex are studied.