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 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.