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

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 verifying , 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 with K ₂, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if , 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.