A dichotomy result about Hessenberg matrices associated with measures in the unit circle. (26th June 2019) - Escribano, Carmen; Gonzalo, Raquel; Torrano, Emilio
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.