A dichotomy result about Hessenberg matrices associated with measures in the unit circle

Escribano Iglesias, M. del Carmen and Gonzalo Palomar, Raquel and Torrano Gimenez, Emilio (2019). A dichotomy result about Hessenberg matrices associated with measures in the unit circle. "Mathematical Methods in the Applied Sciences", v. 42 (n. 17); pp. 5845-5855. ISSN 0170-4214. https://doi.org/10.1002/mma.5716.

Description

Title: A dichotomy result about Hessenberg matrices associated with measures in the unit circle
Author/s:
  • Escribano Iglesias, M. del Carmen
  • Gonzalo Palomar, Raquel
  • Torrano Gimenez, Emilio
Item Type: Article
Título de Revista/Publicación: Mathematical Methods in the Applied Sciences
Date: 2019
ISSN: 0170-4214
Volume: 42
Subjects:
Freetext Keywords: Hermitian positive matrices, Compact perturbation, Hessenberg matrices, Orthogonal polynomial
Faculty: E.T.S. de Ingenieros Informáticos (UPM)
Department: Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients urn:x wiley:mma:media:mma5716:mma5716-math-0001 verifying urn:x-wiley:mma:media:mma5716:mma5716-math-0002, ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D=SR+K2 with K2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that urn:x-wiley:mma:media:mma5716:mma5716-math-0003 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if urn:x-wiley:mma:media:mma5716:mma5716-math-0004, then D=SR+Kp with Kp an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle.

Funding Projects

TypeCodeAcronymLeaderTitle
Government of SpainMTM2016-80582-RUnspecifiedUniversidad Politécnica de MadridTopología digital y matrices infinitas: aplicaciones al procesamiento de imágenes digitales

More information

Item ID: 67092
DC Identifier: https://oa.upm.es/67092/
OAI Identifier: oai:oa.upm.es:67092
DOI: 10.1002/mma.5716
Official URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.5716
Deposited by: Memoria Investigacion
Deposited on: 17 May 2021 07:57
Last Modified: 17 May 2021 07:57
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