A dichotomy result about Hessenberg matrices associated with measures in the unit circle. (26th June 2019)

• Record Type: Journal Article
• Authors: Escribano, Carmen; Gonzalo, Raquel; Torrano, Emilio
• Other Names: Vigo-Aguiar Jesus guestEditor.; Kumam Poom guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 17(2019)
• Page Start: 5845
• Page End: 5855
• Publication Date: 2019-06-26
• Subjects: compact perturbation -- Hermitian positive matrices -- Hessenberg matrix -- orthogonal polynomial; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5716
• Languages: English
• ISSNs: 0170-4214

Abstract

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 { α n } n = 0 ∞ verifying ∑ n = 0 ∞ | α n | 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 =S R +K 2 with K 2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that D = U ∗ Λ U + K 2 with K 2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if ∑ n = 0 ∞ | α n | p < ∞, then D =S R +K p with K p an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle.