Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts
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| Format: | Preprint |
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2024
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| _version_ | 1866918490077659136 |
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| author | Benhida, Chafiq Curto, Raúl E. Exner, George R. |
| author_facet | Benhida, Chafiq Curto, Raúl E. Exner, George R. |
| contents | 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 Lévy-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_15000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts Benhida, Chafiq Curto, Raúl E. Exner, George R. Functional Analysis 47B20, 47B37 (Primary) 44A60 (Secondary) 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 Lévy-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. |
| title | Signed representing measures (Berger-type charges) in subnormality and related properties of weighted shifts |
| topic | Functional Analysis 47B20, 47B37 (Primary) 44A60 (Secondary) |
| url | https://arxiv.org/abs/2405.15000 |