Maximal norm Hankel operators

A Hankel operator H-phi on the Hardy space H-2 of the unit circle with analytic symbol.has minimal norm if parallel to H-phi parallel to = parallel to phi parallel to(2) and maximal norm if parallel to H-phi parallel to = parallel to phi parallel to(infinity). The Hankel operator H-phi has both minimal and maximal norm if and only if vertical bar phi vertical bar is constant almost everywhere on the unit circle or, equivalently, if and only if phi is a constant multiple of an inner function. We show that if H.is norm-attaining and has maximal norm, then H-phi has minimal norm. If vertical bar phi vertical bar is continuous but not constant, then H-phi has maximal norm if and only if the set at which vertical bar phi vertical bar = parallel to phi parallel to(infinity) has nonempty intersection with the spectrum of the inner factor of phi. We obtain further results illustrating that the case of maximal norm is in general related to irregular behavior of log vertical bar phi vertical bar or the argument of.near a maximum point of vertical bar phi vertical bar. The role of certain positive functions coined apical Helson-Szego weights is discussed in the former context. (c) 2023 The Author(s). Published by Elsevier Inc.

Automated summary

This paper studies the Hankel operator on the Hardy space and discusses its minimal and maximal norms, as well as the relationship between the maximal norm and the properties of the function.