Conditional positive definiteness as abridge between k-hyponormality and n-contractivity

Chafiq Benhida; Raul E Curto; George R Exner

doi:10.1016/j.laa.2021.05.004

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Conditional positive definiteness as abridge between k-hyponormality and n-contractivity

Journal article

Peer reviewed

Conditional positive definiteness as abridge between k-hyponormality and n-contractivity

Chafiq Benhida, Raul E Curto and George R Exner

Linear algebra and its applications, Vol.625, pp.146-170

09/15/2021

DOI: 10.1016/j.laa.2021.05.004

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https://arxiv.org/pdf/2012.10962View

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Abstract

For sequences alpha {alpha(n)}(n=0)(infinity) of positive real numbers, called weights, we study the weighted shift operators W-alpha having the property of moment infinite divisibility (MID); that is, for any p > 0, the Schur power W-alpha(p) is subnormal. We first prove that W-alpha is MID if and only if certain infinite matrices log M-gamma(0) and log M-gamma(1) are conditionally positive definite (CPD). Here gamma is the sequence of moments associated with alpha, M-gamma(0), M-gamma(1) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of W-alpha, and log is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between k-hyponormality and n-contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift W-alpha is MID if and only if for all p > 0, M-gamma(p)(0) and M-gamma(p)(1) are CPD. (C) 2021 Elsevier Inc. All rights reserved.

Mathematics

Mathematics, Applied

Physical Sciences

Science & Technology

Details

Title: Subtitle: Conditional positive definiteness as abridge between k-hyponormality and n-contractivity
Creators: Chafiq Benhida - UFR de Mathématiques, Université des Sciences et Technologies de Lille, F-59655 Villeneuve-d'Ascq Cedex, France
Raul E Curto - University of Iowa
George R Exner - Bucknell University
Resource Type: Journal article
Publication Details: Linear algebra and its applications, Vol.625, pp.146-170
DOI: 10.1016/j.laa.2021.05.004
ISSN: 0024-3795
eISSN: 1873-1856
Publisher: ELSEVIER SCIENCE INC
Number of pages: 25
Grant note: DOI: 10.13039/100008893, name: University of Iowa; DOI: 10.13039/501100007856, name: Université Lille 1 - Sciences et Technologies
Language: English
Date published: 09/15/2021
Academic Unit: Mathematics
Record Identifier: 9984240768102771

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