Minimal norm Hankel operators

TL;DR

This paper studies minimal norm Hankel operators in Hardy spaces, characterizing when they attain the lower norm bound and exploring classes of symbols that generate such operators, extending known results to higher dimensions.

Contribution

The paper identifies classes of symbols producing minimal norm Hankel operators and refines existing counter-examples, advancing understanding beyond the one-dimensional case.

Findings

Minimal norm Hankel operators characterized by inner functions in 1D.

Existence of additional symbols generating minimal norm operators in higher dimensions.

Refined counter-examples illustrating the complexity of the problem.

Abstract

Let $\varphi$ be a function in the Hardy space $H^2(\mathbb{T}^d)$. The associated (small) Hankel operator $\mathbf{H}_\varphi$ is said to have minimal norm if the general lower norm bound $\|\mathbf{H}_\varphi\| \geq \|\varphi\|_{H^2(\mathbb{T}^d)}$ is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If $d=1$, then $\mathbf{H}_\varphi$ has minimal norm if and only if $\varphi$ is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when $d\geq2$, but in this case there are other possibilities as well. We investigate two different classes of symbols generating minimal norm Hankel operators and obtain two different refinements of a counter-example due to Ortega-Cerd\`{a} and Seip.