Unbounded subnormal weighted shifts on directed trees I

TL;DR

This paper introduces a novel approximation-based method to verify subnormality of unbounded weighted shifts on directed trees, establishing new sufficient conditions and extending Lambert's characterization to unbounded cases with quasi-analytic vectors.

Contribution

It presents a new approximation technique for subnormality, develops sufficient conditions, and extends Lambert's characterization to unbounded weighted shifts on directed trees.

Findings

Established diverse sufficient conditions for subnormality.

Invented an approach via consistent systems of probability measures.

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.