Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

Tomeo, V.; Torrano, Emilio

doi:10.1016/j.laa.2011.04.027

Cited by 4 publications

(1 citation statement)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is a subnormal operator if it is the restriction of a normal operator on an invariant subspace. From the results in Atzmon, Conway, and Tomeo and Torrano,…”

Section: Asymptotics Of Hessenberg Matrices Associated With Toeplitz mentioning

confidence: 99%

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

Escribano

Gonzalo

Torrano

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

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 false{αnfalse}n=0∞ verifying ∑n=0∞false|αn|2<∞, 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 boldD=U∗normalΛboldU+K2 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if ∑n=0∞false|αn|p<∞, 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.

show abstract

“…It is a subnormal operator if it is the restriction of a normal operator on an invariant subspace. From the results in Atzmon, Conway, and Tomeo and Torrano,…”

Section: Asymptotics Of Hessenberg Matrices Associated With Toeplitz mentioning

confidence: 99%