Maximal norm Hankel operators

TL;DR

This paper characterizes when Hankel operators on the Hardy space have maximal or minimal norm, linking these cases to properties of the symbol function, especially its magnitude and inner factor, and explores irregular behaviors affecting the norm.

Contribution

It provides a complete characterization of maximal and minimal norm Hankel operators in terms of the symbol's properties, including continuity and inner factor behavior.

Findings

Hankel operators have both maximal and minimal norm only if the symbol's magnitude is constant almost everywhere.

Maximal norm Hankel operators are related to the set where the symbol attains its maximum intersecting the spectrum of its inner factor.

Irregularities in the logarithm of the symbol's magnitude influence the operator's norm behavior.

Abstract

A Hankel operator $\mathbf{H}_\varphi$ on the Hardy space $H^2$ of the unit circle with analytic symbol $\varphi$ has minimal norm if $\|\mathbf{H}_\varphi\|=\|\varphi \|_2$ and maximal norm if $\|\mathbf{H}_\varphi\| = \|\varphi\|_\infty$. The Hankel operator $\mathbf{H}_\varphi$ has both minimal and maximal norm if and only if $|\varphi|$ is constant almost everywhere on the unit circle or, equivalently, if and only if $\varphi$ is a constant multiple of an inner function. We show that if $\mathbf{H}_\varphi$ is norm-attaining and has maximal norm, then $\mathbf{H}_\varphi$ has minimal norm. If $|\varphi|$ is continuous but not constant, then $\mathbf{H}_\varphi$ has maximal norm if and only if the set at which $|\varphi|=\|\varphi\|_{\infty}$ has nonempty intersection with the spectrum of the inner factor of $\varphi$. We obtain further results illustrating that the case of maximal norm is in general related to "irregular" behavior of $\log |\varphi|$ or the argument of $\varphi$ near a "maximum point" of $|\varphi|$. The role of certain positive functions coined apical Helson--Szeg\H{o} weights is discussed in the former context.