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arXiv · 2405.15000

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

Chafiq Benhida·Raúl E. Curto·George R. Exner
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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.

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Chafiq Benhida, Raúl E. Curto, George R. Exner. 2024-05-23. Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts. https://doi.org/10.1007/s00025-025-02574-4

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