Maximal norm Hankel operators

A Hankel operator ${\mathbf{H}}_{\phi }$ on the Hardy space $H^2$ of the unit circle with analytic symbol $\phi$ has minimal norm if $\mathrm{\parallel }{\mathbf{H}}_{\phi }\mathrm{\parallel }=\mathrm{\parallel }\phi {\mathrm{\parallel }}_2$ and maximal norm if $\mathrm{\parallel }{\mathbf{H}}_{\phi }\mathrm{\parallel }=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$. The Hankel operator ${\mathbf{H}}_{\phi }$ has both minimal and maximal norm if and only if $\mathrm{\mid }\phi \mathrm{\mid }$ 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 ${\mathbf{H}}_{\phi }$ is norm-attaining and has maximal norm, then ${\mathbf{H}}_{\phi }$ has minimal norm. If $\mathrm{\mid }\phi \mathrm{\mid }$ is continuous but not constant, then ${\mathbf{H}}_{\phi }$ has maximal norm if and only if the set at which $\mathrm{\mid }\phi \mathrm{\mid }=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$ 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 $\log \mathrm{\mid }\phi \mathrm{\mid }$ or the argument of $\phi$ near a “maximum point” of $\mathrm{\mid }\phi \mathrm{\mid }$. The role of certain positive functions coined apical Helson–Szegő weights is discussed in the former context.