Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts

TL;DR

This paper develops a general theory of Berger-type charges for weighted shifts, revealing new measure-related properties from k-hyponormality and hyperexpansivity, and exploring their implications for subnormality and positivity.

Contribution

It introduces a comprehensive framework for Berger-type charges, linking k-hyponormality to measure positivity and providing new representations for hyperexpansive shifts.

Findings

k-hyponormality implies positivity of certain measure densities

Berger-type charges can represent hyperexpansive shifts

non-subnormal shifts may be scaled to be conditionally positive definite

Abstract

In the study of the geometrically regular weighted shifts (GRWS) -- see [5] -- signed power representing measures (which we call Berger-type charges) played an important role. Motivated by their utility in that context, we establish a general theory for Berger-type charges. We give the first result of which we are aware showing that k-hyponormality alone (as opposed to subnormality) yields measure/charge-related information. More precisely, for signed countably atomic measures with a decreasing sequence of atoms we prove that k-hyponormality of the associated shift forces positivity of the densities of the largest k+1 atoms. Further, for certain completely hyperexpansive weighed shifts, we exhibit a Berger-type charge representation, in contrast (but related) to the classical L\'{e}vy-Khinchin representation. We use Berger-type charges to investigate when a non-subnormal GRWS weighted shift may be scaled to become conditionally positive definite, and close with an example indicating a distinction between the study of moment sequences and the study of weighted shifts.