Maximal norm Hankel operators

Abstract

A Hankel operator Hφ on the Hardy space H2 of the unit circle with analytic symbol φ has minimal norm if ‖Hφ‖=‖φ‖2 and maximal norm if ‖Hφ‖=‖φ‖∞. The Hankel operator Hφ has both minimal and maximal norm if and only if |φ| is constant almost everywhere on the unit circle or, equivalently, if and only if φ is a constant multiple of an inner function. We show that if Hφ is norm-attaining and has maximal norm, then Hφ has minimal norm. If |φ| is continuous but not constant, then Hφ has maximal norm if and only if the set at which |φ|=‖φ‖∞ has nonempty intersection with the spectrum of the inner factor of φ. We obtain further results illustrating that the case of maximal norm is in general related to “irregular” behavior of log⁡|φ| or the argument of φ near a “maximum point” of |φ|. The role of certain positive functions coined apical Helson–Szegő weights is discussed in the former context.