Maximal norm Hankel operators

Abstract

A Hankel operator $\mathbf{H}_φ$ on the Hardy space $H^2$ of the unit circle with analytic symbol $φ$ has minimal norm if $\|\mathbf{H}_φ\|=\|φ\|_2$ and maximal norm if $\|\mathbf{H}_φ\| = \|φ\|_\infty$. The Hankel operator $\mathbf{H}_φ$ 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 $\mathbf{H}_φ$ is norm-attaining and has maximal norm, then $\mathbf{H}_φ$ has minimal norm. If $|φ|$ is continuous but not constant, then $\mathbf{H}_φ$ has maximal norm if and only if the set at which $|φ|=\|φ\|_{\infty}$ 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.